Adaptive digital predistortion for wideband high crest ...

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Laki, Bradley Dean (2013) Adaptive digital predistortion

for wideband high crest factor applications based on the

WACP optimization objective. PhD thesis, James Cook

University.

Access to this file is available from:

http://researchonline.jcu.edu.au/40267/

The author has certified to JCU that they have made a reasonable effort to gain

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ResearchOnline@JCU

Adaptive Digital Predistortion For Wideband

High Crest Factor Applications Based On The

WACP Optimization Objective

Thesis submitted by

Bradley Dean LAKI, BE (Hons) QUT, MSc UA

in July 2013

for the degree of Doctor of Philosophy

in the School of Engineering & Physical Sciences

James Cook University

Statement On The Contribution

Of Others

The following people deserve special mention for their contribution to this research.

Principal supervisor Cornelis Jan Kikkert for his research advice, laboratory provi-

sion and manuscript proof reading. Workshop manager John Renehan for his supply

of electrical test equipment and components. Radio technicians David Henry and

Rod Prior (Broadcast Australia), Gary Lane (LaneComm Communications) and

Ian Smart (Channel 7) for their transmitter site access and data collection efforts.

Finally, Jasper Taylor and UniQuest for their assistance in patent filing and commer-

cialization. This research was made possible via a Commonwealth APA scholarship.

i

Acknowledgments

I wish to thank the following people for their support throughout this journey. First

and foremost my family for their unconditional love. My supervisors Cornelis Jan

Kikkert and Owen Kenny for their guidance. Jeff Dale and Graham Weinberg for

their humor and down to earth nature. Craig and Andrea McPherson for too many

things to list here. Thanks guys! Phil Turner for giving me the opportunity to

lecture and last but not least, Greg Borger from Ergon Energy for his support

whilst preparing this final manuscript.

ii

Abstract

Modern digital communication systems utilize OFDM and DS-CDMA signals be-

cause of their superior spectral efficiency and multiple access properties respec-

tively. Examples of such systems include digital television (DVB-T), digital radio

(DAB) and 3rd Generation mobile (WCDMA). A downside to using these signals

however is the need to accommodate their very high Crest Factor, also known as

Peak-to-Average Power Ratio, during transmission. Compared to other communica-

tion signals transmitted with the same average power, these signals generate greater

transmitter distortion since their larger signal peaks drive the transmitter power

amplifier into regions of greater nonlinearity.

Reducing this Crest Factor related distortion, whilst concurrently maintaining

power amplifier efficiency, requires the power amplifier’s transfer characteristic to be

externally linearized. For OFDM and DS-CDMA signals, digital predistortion is the

favored linearization technique given its cost effectiveness, superior signal processing

capability, potential to adapt and ability to linearize the entire transmitter.

In this research thesis, a new digital predistortion technique is proposed. By em-

ploying frequency-domain information feedback, rather than traditional time-domain

signal feedback, the new technique avoids bandwidth limitations in the feedback path

and is hence more suited to wideband applications than current generation tech-

niques. This frequency-domain information feedback is based on the novel Weighted

Adjacent Channel Power (WACP) linearization objective. By incorporating fre-

quency dependent weighting into the standard accumulation of Adjacent Channel

Power (ACP), this novel linearization objective is able to discriminate between spec-

tral distortion components and hence control the location of spectral distortion re-

duction. This makes linearization more robust in the presence of residual predistor-

tion filtering inaccuracies.

To accommodate the power amplifier memory effects associated with wideband

signal modulation, the new technique uses a predistortion filter architecture based on

the classic nonlinear dynamic Volterra Series. Mindful of Volterra kernel efficiency,

the technique further applies a novel hybridized triple-stage pruning strategy, leaving

the kernel size not only linear with respect to memory, but also independent of

iii

hardware sampling rate implementation. This pruning activity ultimately reduces

the number of dynamic filter parameters needing to be estimated.

This new digital predistortion technique utilizes generic mathematical optimiza-

tion to estimate the resulting predistortion filter kernel, with frequency-domain

WACP interpreted as the linearizing optimization objective. In line with theory,

the objective is assumed to be nonconvex and hence both global and local opti-

mization algorithms are employed to achieve true global convergence. This is in

direct contrast to the traditional and competing Direct Learning technique which is

more focused on quick local convergence, and less focused on guaranteeing maximum

linearization performance corresponding to the objective global minimum.

Since predistortion filter kernel estimation is performed using the transmitted

wideband signal, the new technique is ultimately on-air adaptive. This means any

transmitter using the technique will be both on-air and optimally linearized for its

entire operational life.

iv

Contents

Statement On The Contribution Of Others i

Acknowledgments ii

Abstract iii

List of Tables ix

List of Figures xi

Acronyms xii

Mathematical Symbols xvi

1 Introduction 1

1.1 The Need For Transmitter Linearization . . . . . . . . . . . . . . . . . . 3

1.2 Digital Predistortion Linearization . . . . . . . . . . . . . . . . . . . . . 8

1.3 Conclusions Reached . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Literature Review of Digital Predistortion 9

2.1 Predistortion Filter Architectures . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Data Predistortion (Narrowband) . . . . . . . . . . . . . . . . . 12

2.1.2 Signal Predistortion (Narrowband) . . . . . . . . . . . . . . . . . 12

2.1.3 Volterra Series (Wideband) . . . . . . . . . . . . . . . . . . . . . 14

2.1.4 Memory Polynomial (Wideband) . . . . . . . . . . . . . . . . . . 15

2.1.5 NARMA Filter (Wideband) . . . . . . . . . . . . . . . . . . . . . 16

2.1.6 Hammerstein and Wiener Filters (Wideband) . . . . . . . . . . 16

2.1.7 TNTB Model (Wideband) . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Predistortion Filter Parameter Estimation . . . . . . . . . . . . . . . . . 18

2.2.1 Model Based Derivation (Narrowband /Wideband) . . . . . . . 19

2.2.2 Indirect Learning (Wideband) . . . . . . . . . . . . . . . . . . . . 21

2.2.3 Direct Learning (Wideband) . . . . . . . . . . . . . . . . . . . . . 22

v

2.2.4 Spectral Power Feedback Learning (Wideband) . . . . . . . . . 24

2.3 Opportunity For Further Beneficial Research . . . . . . . . . . . . . . . 25

3 Research Scope & Outputs 28

3.1 Statement of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Target Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 Delimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.5 Commercialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.6 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Laboratory Transmitter Testbed 32

4.1 IQ Modulation Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Vector Signal Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.3 Driver Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4 Power Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.5 Spectrum Analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.6 Signal Encoding and Modulation . . . . . . . . . . . . . . . . . . . . . . 38

4.7 Predistortion Filtering, Objective Function Measurement and

Mathematical Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.8 Console Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Volterra Series Modeling of Amplifier & Predistorter 43

5.1 Introduction To The Volterra Series . . . . . . . . . . . . . . . . . . . . 44

5.2 RF and Baseband Transmitter Models . . . . . . . . . . . . . . . . . . . 47

5.3 Predistortion Filter Architecture . . . . . . . . . . . . . . . . . . . . . . 52

6 Digital Predistortion In The Time-Domain 54

6.1 Intuitive Graphical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.2 Mathematical Operator Analysis . . . . . . . . . . . . . . . . . . . . . . 56

6.3 Ideal Predistorter Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.4 Theoretical Limits of Digital Predistortion . . . . . . . . . . . . . . . . 61

7 Digital Predistortion In The Frequency-Domain 62

8 Maximum Order Of Predistorter Nonlinearity 70

9 SPFL Strategy With New WACP Optimization Objective 72

9.1 Mathematical Framework of SPFL Strategy . . . . . . . . . . . . . . . . 72

9.2 New WACP Optimization Objective . . . . . . . . . . . . . . . . . . . . 74

vi

10 Mathematical Optimization Process 80

10.1 Two Distinct Phases of Optimization . . . . . . . . . . . . . . . . . . . . 80

10.2 Maximal Convergence Reliability . . . . . . . . . . . . . . . . . . . . . . 81

10.3 Optimization Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

10.4 Initial Setting Optimization Schedule . . . . . . . . . . . . . . . . . . . . 84

10.5 On-Air Adaption Optimization Schedule . . . . . . . . . . . . . . . . . . 86

11 Mathematical Optimization Algorithms 90

11.1 Algorithm Classifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

11.2 Theoretical Shortlist of Algorithms . . . . . . . . . . . . . . . . . . . . . 93

11.3 Practical Suitability of Shortlisted Algorithms . . . . . . . . . . . . . . 94

11.4 Selected Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . 99

11.5 Refined Optimization Schedules . . . . . . . . . . . . . . . . . . . . . . . 100

12 Pruning The Predistorter Volterra Kernel 101

12.1 The Pruning Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

12.2 Memory Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

12.3 Refined Optimization Vector Space . . . . . . . . . . . . . . . . . . . . . 111

13 Performance Baseline 113

13.1 Research Journals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

13.2 Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

13.3 Application Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

13.4 Transmitter Manufacturers . . . . . . . . . . . . . . . . . . . . . . . . . . 117

13.5 Transmitter Operating Manuals . . . . . . . . . . . . . . . . . . . . . . . 117

13.6 Transmission Service Providers . . . . . . . . . . . . . . . . . . . . . . . 117

13.7 Derivation of Performance Baseline . . . . . . . . . . . . . . . . . . . . . 122

14 Performance Of The Proposed Predistortion Technique 124

14.1 Initial Setting Performance Testing . . . . . . . . . . . . . . . . . . . . . 124

14.2 On-Air Adaption Performance Testing . . . . . . . . . . . . . . . . . . . 131

14.3 Crest Factor Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

15 Summary & Conclusion 140

Bibliography 143

A Shortlisted Optimization Algorithms 170

A.1 Gradient Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

A.2 Trust Region Newton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

A.3 Alpha Branch & Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

vii

A.4 Nelder-Mead Simplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

A.5 Genetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

B Mathematical Derivations & Formulas 206

B.1 Estimation of the Gradient Vector g . . . . . . . . . . . . . . . . . . . . 206

B.1.1 Finite Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

B.1.2 Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

B.2 Estimation of the Hessian Matrix H . . . . . . . . . . . . . . . . . . . . 210

B.2.1 Finite Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

B.2.2 Hybridization of Finite Differences & Least Squares . . . . . . 211

B.3 Eigen Properties of the Hessian Matrix H . . . . . . . . . . . . . . . . . 213

C Patent 217

D Publications 222

E Data Sheets 243

viii

List of Tables

1.1 Comparison of Crest Factors for various signal modulations . . . . . . 7

10.1 Simplest ascending nonlinear order optimization schedule . . . . . . . 83

10.2 Summarizing steps of proposed optimization schedule . . . . . . . . . . 84

10.3 Proposed Initial Setting optimization schedule . . . . . . . . . . . . . . 86

10.4 Proposed On-Air Adaption optimization schedule . . . . . . . . . . . . 88

11.1 Theoretical shortlist of optimization algorithms . . . . . . . . . . . . . 93

11.2 Finalized Initial Setting optimization schedule . . . . . . . . . . . . . . 100

11.3 Finalized On-Air Adaption optimization schedule . . . . . . . . . . . . 100

12.1 R-sample delay increments . . . . . . . . . . . . . . . . . . . . . . . . . . 105

12.2 Predistortion filter memory estimates . . . . . . . . . . . . . . . . . . . . 110

12.3 Maximum optimization load . . . . . . . . . . . . . . . . . . . . . . . . . 112

13.1 Wideband, hardware tested performance in research papers . . . . . . 115

13.2 Summary of wideband, hardware tested, adaptive performance . . . . 122

14.1 Initial Setting optimization schedule . . . . . . . . . . . . . . . . . . . . 125

14.2 Initial Setting convergence rate properties . . . . . . . . . . . . . . . . . 130

14.3 On-Air Adaption optimization schedule . . . . . . . . . . . . . . . . . . 131

14.4 Predistortion filter kernel coefficients for DVB-T . . . . . . . . . . . . . 135

14.5 Predistortion filter kernel coefficients for WCDMA . . . . . . . . . . . . 136

14.6 Predistortion filter kernel coefficients for DAB . . . . . . . . . . . . . . 137

A.1 Objective function measurements per geometric update . . . . . . . . . 197

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List of Figures

1.1 Summary of proposed digital predistortion technique . . . . . . . . . . 2

1.2 Conventional radio transmitter architecture . . . . . . . . . . . . . . . . 4

1.3 Architecture of power amplifier module . . . . . . . . . . . . . . . . . . 4

1.4 Spectral regrowth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Placement of digital predistortion filter . . . . . . . . . . . . . . . . . . . 8

2.1 Hammerstein, Wiener and Augmented filter architectures . . . . . . . 17

2.2 Twin Nonlinear Two-Box (TNTB) model architectures . . . . . . . . . 18

2.3 Model Based Derivation (MBD) strategy . . . . . . . . . . . . . . . . . 19

2.4 Indirect Learning strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Direct Learning strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.6 Spectral Power Feedback Learning (SPFL) strategy . . . . . . . . . . . 25

2.7 Summary of literature review . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1 Block diagram of the laboratory transmitter testbed . . . . . . . . . . 33

4.2 Photo of the laboratory transmitter testbed . . . . . . . . . . . . . . . . 34

4.3 Complementary Cumulative Distribution Functions . . . . . . . . . . . 41

5.1 Operator form of Volterra Series . . . . . . . . . . . . . . . . . . . . . . . 45

5.2 RF transmitter model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3 Baseband transmitter model . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.4 Baseband transmitter model with predistortion filter inserted . . . . . 52

6.1 Graphical analysis of digital predistortion in the time-domain . . . . . 55

6.2 Predistorter-Amplifier cascade and Distortion Array . . . . . . . . . . 57

6.3 Parasitic components generated by Pm[ ⋅ ] . . . . . . . . . . . . . . . . 58

6.4 Equivalent cascade nonlinearity Q[ ⋅ ] . . . . . . . . . . . . . . . . . . . . 60

7.1 Equivalent cascade nonlinearity Q[ ⋅ ] . . . . . . . . . . . . . . . . . . . . 63

7.2 Power spectra Sqlql(f) prior to predistortion . . . . . . . . . . . . . . . 64

7.3 Behavior of power spectra after 3rd order predistortion . . . . . . . . . 65

7.4 Power spectra before and after 3rd order predistortion . . . . . . . . . . 66

x

7.5 Power spectra after 5th order predistortion . . . . . . . . . . . . . . . . 67

7.6 Equivalent cascade and predistorter output power spectra . . . . . . . 69

8.1 Dominant elements of the Distortion Array . . . . . . . . . . . . . . . . 71

9.1 Parameter estimation based on the SPFL strategy . . . . . . . . . . . . 73

9.2 Output power spectra after predistortion using ACP objective . . . . 75

9.3 Linear, quadratic and higher order weighting functions . . . . . . . . . 76

9.4 Output power spectra for over- and under-weighting scenarios . . . . . 77

10.1 Performance of Initial Setting optimization schedule . . . . . . . . . . . 87

10.2 Power spectra for progressively larger forced drifts . . . . . . . . . . . . 87

12.1 WCDMA memory estimation probing . . . . . . . . . . . . . . . . . . . 109

12.2 DVB-T memory estimation probing . . . . . . . . . . . . . . . . . . . . . 109

13.1 ACD Shoulder Height and Attenuation . . . . . . . . . . . . . . . . . . . 114

13.2 Channel 7 Yarrawonga amplifier output . . . . . . . . . . . . . . . . . . 118

13.3 ABC Mt Stuart transmission . . . . . . . . . . . . . . . . . . . . . . . . . 120

13.4 SBS Mt Stuart transmission . . . . . . . . . . . . . . . . . . . . . . . . . 121

14.1 Initial Setting testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

14.2 Initial Setting testing with inband power temporarily removed . . . . 126

14.3 First stage of On-Air Adaption testing . . . . . . . . . . . . . . . . . . . 133

14.4 Second stage of On-Air Adaption testing . . . . . . . . . . . . . . . . . 133

14.5 CCDF of the predistortion filter output signal . . . . . . . . . . . . . . 138

A.1 Flowchart of Gradient Descent optimization algorithm . . . . . . . . . 173

A.2 Flowchart of Trust Region Newton optimization algorithm . . . . . . . 183

A.3 Fathoming example for the ABB optimization algorithm . . . . . . . . 191

A.4 Branching example for the ABB optimization algorithm . . . . . . . . 192

A.5 Flowchart of Alpha Branch & Bound optimization algorithm . . . . . 194

A.6 Flowchart of Nelder-Mead Simplex optimization algorithm . . . . . . . 199

A.7 Overview of Genetic optimization algorithm . . . . . . . . . . . . . . . . 200

A.8 Biological evolutionary process used to refine Population Pool . . . . . 202

A.9 Gene plots of the Population Pool . . . . . . . . . . . . . . . . . . . . . . 204

A.10 Flowchart of Genetic optimization algorithm . . . . . . . . . . . . . . . 205

xi

Acronyms

3G 3rd Generation

ABB Alpha Branch & Bound

ABC Australian Broadcasting Corporation

ACD Adjacent Channel Distortion

ACMA Australian Communications and Media Authority

ACP Adjacent Channel Power

ACPR Adjacent Channel Power Ratio

ADC Analog-to-Digital Converter

a.k.a. Also Known As

AM-AM Amplitude Modulation to Amplitude Modulation

AM-PM Amplitude Modulation to Phase Modulation

BER Bit Error Rate

BW Bandwidth

CCD Co-Channel Distortion

CCDF Complementary Cumulative Distribution Function

CCP Co-Channel Power

CCPR Co-Channel Power Ratio

CDMA Code Division Multiple Access

CF Crest Factor or Center Frequency

CM Counter-Measure

xii

CPM Continuous Phase Modulation

DAB Digital Audio Broadcasting

DAC Digital-to-Analog Converter

dB deciBel

dBc deciBel relative to Carrier

dBm deciBel relative to milli Watt

DDR Dynamic Deviation Reduction

DOS Disk Operating System

DS-CDMA Direct Sequence Code Division Multiple Access

DSP Digital Signal Processor

DVB-T Digital Video Broadcasting - Terrestrial

EE&R Envelope Elimination & Restoration

EIRP Effective Isotropic Radiated Power

ETSI European Telecommunications Standards Institute

EVM Error Vector Magnitude

FCC Federal Communications Commission

FET Field Effect Transistor

FIR Finite Impulse Response

FM Frequency Modulation

FPGA Field Programmable Gate Array

GaAs Gallium Arsenide

GD Gradient Descent

GPIB General Purpose Interface Bus

Hz Hertz

IDE Integrated Development Environment

IEEE Institute of Electrical and Electronics Engineers

xiii

IF Intermediate Frequency

IFFT Inverse Fast Fourier Transform

IIR Infinite Impulse Response

IMT-2000 International Mobile Telecommunications - 2000

IP Intellectual Property

IQ Inphase Quadrature

KW Kilo Watt

LAC Lower Adjacent Channel

LDMOS Laterally Diffused Metal Oxide Semiconductor

LINC LInear amplification using Nonlinear Components

LMS Least Mean Squares

LTI Linear Time Invariant

LU Lower Upper

LUT Look Up Table

MER Modulation Error Ratio

MHz Mega Hertz

MSK Minimum Shift Keying

NEC Nippon Electric Company

NMA Nonlinear Moving Average

NMS Nelder-Mead Simplex

OBO Output Back Off

OFDM Orthogonal Frequency Division Multiplexing

PA Power Amplifier

PAPR Peak-to-Average Power Ratio

PC Personal Computer

PDF Probability Density Function

xiv

PSD Power Spectral Density

PSK Phase Shift Keying

QAM Quadrature Amplitude Modulation

QPSK Quadrature Phase Shift Keying

RAM Random Access Memory

RBW Resolution BandWidth

R&D Research & Development

RF Radio Frequency

RLS Recursive Least Squares

RRC Root Raised Cosine

SBS Special Broadcasting Service

SPFL Spectral Power Feedback Learning

SR1 Symmetric-Rank-1

SWT SWeep Time

TM-I Transmission Mode - I

TNTB Twin Nonlinear Two-Box

TRN Trust Region Newton

TWT Travelling Wave Tube

UAC Upper Adjacent Channel

UHF Ultra High Frequency

UMTS Universal Mobile Telecommunications System

V Volt

VBW Volterra Behavioral Wideband or Video BandWidth

vs Verses

W Watt

WACP Weighted Adjacent Channel Power

WCDMA Wideband Code Division Multiple Access

xv

Mathematical Symbols

N.B. Vector parameters are represented by boldface text e.g. h

≜ Defined as

∆ Trust region radius (TRN algorithm)

∆f Discrete frequency step size

∥ ⋅ ∥ Euclidean Norm

⌈ ⋅ ⌉ Ceiling operator

[ ⋅ ]T Vector transposition

αi Variable (ith dimension) ensuring convexity of L(h) (ABB algorithm)

αideali Ideal αi (ABB algorithm)

αpraci Practically computable αi (ABB algorithm)

χ Chromosomes used to estimate initial Population Pool (Genetic algorithm)

ε Small positive scalar used in Finite Difference estimation

κ Number of chromosomes in Mating Pool (Genetic algorithm)

λ General eigenvalue of H

Λ Diagonal spectral matrix of H = QΛQT

µ Step distance (GD algorithm)

µideal Ideal step distance (GD algorithm)

µhkStep distance chosen for iteration hk (GD algorithm)

Φ Diagonal matrix of α values (ABB algorithm)

xvi

σ Scalar parameter of the Symmetric-Rank-1 Update equation

Υ Mean square criterion on which to automate exiting (NMS algorithm)

Υ0 Exit threshold for Υ (NMS algorithm)

� Number of chromosomes in Offspring Pool (Genetic algorithm)

ϑ Chromosome concentrations when refinement is halted (Genetic algorithm)

ξ Number of chromosomes in Population Pool (Genetic algorithm)

A[ ⋅ ] System amplifier operator within baseband transmitter model

A[ ⋅ ] System amplifier operator within RF transmitter model

An[ ⋅ ] nth order amplifier operator within baseband transmitter model

An{ s[k], . . . , s[k] } n-linear amplifier operator within baseband transmitter model

An[ ⋅ ] nth order amplifier operator within RF transmitter model

An{ s1(t), . . . , sn(t)} n-linear amplifier operator within RF transmitter model

a Linear parameter of linear parametric model (Least Squares g estimation)

an(τ1, . . . , τn) nth order amplifier Volterra kernel within baseband transmitter model

an(τ1, . . . , τn) nth order amplifier Volterra kernel within RF transmitter model

B(h) Optimization objective as a function of h

b Value of objective function. B(h) = b

bo Value of objective function at optimization algorithm’s current iterate

bmin Global minimum value of objective function. B(ho) = bmin

B Spectral bandwidth of s(t) (Hertz)

Bc Continuous-time spectral bandwidth (Hertz)

Bd Discrete-time spectral bandwidth (cycles/sample). Bd = Bc/Fs

C Infinite set representing the complex number plane

C Scalar averaging term (Least Squares g estimation)

c Constant parameter of linear parametric model (Least Squares g estimation)

cihkith candidate step distance generated during iteration hk (GD algorithm)

xvii

Di ith Gerschgorin disk (ABB algorithm)

d Small positive increment used to define S0 (NMS algorithm)

d Deviation from optimization algorithm’s current iterate

d Modified d vector (Least Squares g estimation)

ei Unit vector along the ith dimension of the vector space h

f Frequency (Hertz)

fCarrier Carrier frequency

fE Transmission band edge frequency (Hertz)

Fs Sampling frequency (samples/second)

f0 Adjacent channel frequency at which the weighting drops to zero. W (f0) = 0

G Amplifier gain

g Gradient vector

H Hessian matrix

H[ ⋅ ] General system Volterra operator

Hn[ ⋅ ] General nth order Volterra operator

Hn{s1(t), . . . , sn(t)} General n-linear Volterra operator

Hn(f1, . . . , fn) General n-dimensional Fourier Transform of hn(τ1, . . . , τn)

HB(h) Hessian matrix of B(h)

[HB]Rg Rg’s objective function interval Hessian matrix (ABB algorithm)

¯H Lower interval bound of [HB]Rg

H Upper interval bound of [HB]Rg

HL(h) Hessian matrix of L(h)

hn(τ1, . . . , τn) General nth order Volterra kernel

h Optimization vector space (row vectorized)

hL Lower constraint on the optimization vector space h

hU Upper constraint on the optimization vector space h

xviii

hL,Rg Lower constraint on the general boxed region Rg

hU,Rg Upper constraint on the general boxed region Rg

hk kth iteration of the optimization algorithm

ho Optimal value of h which minimizes the objective function. B(ho) = bmin

hs Vector space starting point of mathematical optimization (NMS algorithm)

hs Scheduled optimization variable vector. hs ⊂ h

I Identity matrix

i1, . . . , im Delay variables as used in pm[i1, . . . , im]

i Delay variable as used in pm[i] and M estimation distortion probing

L Maximum optimization load

LAC Integration domain of the Lower Adjacent Channel frequencies

L(h) Convex function underestimating B(h) over region Rg (ABB algorithm)

M[ ⋅ ] Mask filter operator within RF and baseband transmitter models

M Predistortion filter discrete-time memory length (samples)

Mc Predistorter continuous-time memory length (seconds). Mc = MTs

m(d ) Objective function quadratic model (TRN algorithm)

n Size of the optimization variable vector hs. n = S for the special case hs = h

P Maximum nonlinearity (odd) of predistortion filter

P Vector averaging term (Least Squares g estimation)

P [ ⋅ ] System predistortion filter operator

Pm[ ⋅ ] mth order predistortion filter operator

pm[i1, . . . , im] mth order predistorter Baseband Volterra kernel

p2a+1[i1, . . . , i2a+1] Kernel of compact predistorter representation

p2a+1[i1, . . . , ia] Predistorter kernel after Volterra Behavioral Wideband pruning

p2a+1[i] Predistorter kernel after Dynamic Deviation Reduction pruning

p2a+1[i] Predistorter kernel after R-Sample Delay Increment pruning

xix

pm[i] mth order predistorter kernel after hybrid pruning

pm Set of mth order predistorter kernel coefficients (row vectorized)

pmR Real part of pm

pmI Imaginary part of pm

P (f) Power spectral density as a function of frequency (Watts/Hertz)

q General eigenvector of H

q(t) Output of predistorter-amplifier or equivalent cascade

ql(t) Output of lth order equivalent cascade nonlinearity

q[k] Discrete-time version of q(t)

ql[k] Discrete-time version of ql(t)

Q Orthogonal modal matrix of H = QΛQT

Q[ ⋅ ] System operator of equivalent cascade

Ql[ ⋅ ] lth order operator of equivalent cascade

R Delay increment within predistorter pruned Volterra Series (samples)

R Matrix averaging term (Least Squares g estimation)

R Infinite set representing the real number line

Rg General boxed region of optimization vector space [hL,Rg ≤ h ≤ hU,Rg]

S Size of the optimization vector space h

S0 Initial simplex (NMS algorithm)

Sk Geometrically updated simplex at optimization step k ≥ 1 (NMS algorithm)

Sss(f) Power spectral density of s(t)

Sqlql(f) Power spectral density of ql(t)

s(t) Signal modulator output signal within RF and baseband transmitter models

s[k] Discrete-time version of s(t)

s(t) RF excitation signal within RF transmitter model

s+1(t) Positive bandpass frequency component of s(t)

xx

s−1(t) Negative bandpass frequency component of s(t)

Ts Sampling period (seconds/sample)

UAC Integration domain of the Upper Adjacent Channel frequencies

U l[ s[k] ] Accumulation of lth order parasitic distortion components

u Unit vector representing the line of B(h) steepest descent (GD algorithm)

v Vector parameter of the Symmetric-Rank-1 Update equation

v Centroid of Sk−1’s best n vertices (NMS algorithm)

v1 → vn+1 Ordered vertices of the previous simplex Sk−1 (NMS algorithm)

ve Simplex expansion point (NMS algorithm)

vic Simplex inside contraction point (NMS algorithm)

voc Simplex outside contraction point (NMS algorithm)

vr Simplex reflection point (NMS algorithm)

W (f) Nonnegative weighting as a function of frequency

WE Weighting at the transmission band edge frequency fE

w Parameter estimation vector (Least Squares g estimation)

Yn(f) Resultant nth order spectral regrowth

Yn(f1, . . . , fn) Component nth order spectral regrowth

xxi

Chapter 1

Introduction

Modern digital communication systems which utilize nonconstant envelope, high

Crest Factor signal modulations such as OFDM and DS-CDMA, must linearize their

power amplifiers in order to achieve power efficient transmission. Of all linearization

techniques, digital predistortion is considered the most promising given its cost

effectiveness and superior signal processing capability.

Predistortion linearization is achieved by inserting a nonlinear digital filter at the

output of the signal modulator and tuning its transfer characteristic to be the inverse

of all downstream transmitter nonlinearities. As a result, the design of any digital

predistortion system can be framed by two fundamental questions. Firstly, what

architecture will the predistortion filter take? Secondly, how will the characterizing

parameters of this filter be estimated? In line with these two design questions, this

thesis presents a digital predistortion technique, novel in two primary respects:

The predistortion filter architecture is based on the dynamic Volterra Series,

judiciously pruned to achieve a kernel size linear with respect to memory,

without adversely affecting linearization potential. This is in direct contrast

to traditional dynamic filter architectures whose kernel size remains exponen-

tial with respect to memory. This change drastically reduces the number of

dynamic filter parameters needing to be estimated, making it far more robust

and suited to wideband applications than current filter architectures.

The predistortion filter parameter estimation strategy is underpinned by

frequency-domain information feedback and generic single objective mathe-

matical optimization. This is in direct contrast to traditional predistortion

techniques which utilize time-domain signal feedback and linear regression op-

timization. This change eliminates bandwidth limitations in the feedback path

and ensures global optimization convergence, ultimately making the technique

far more robust and suited to wideband applications than current generation

techniques.

1

CHAPTER 1. INTRODUCTION 2

Figure 1.1 summarizes the proposed digital predistortion technique, highlighting its

primary wideband novelties in yellow and green respectively. Standard transmitter

elements are represented in blue.

signalmodulator

wideband digitalpredistortion

filter

data in DACs,reconstruction

filters

IQ modulator,frequency

upconverter

poweramplifier

tx out

genericsingle objectivemathematicaloptimization

fewer dynamicparameters to be

estimated

global optimization convergence,wideband frequency-domain

information feedback

WACPoptimization

objective

power spectral densitymeasurements in the

adjacent channel

Figure 1.1: Summary of proposed digital predistortion technique

It is seen here that predistortion filter parameter estimation is modeled as a generic

single objective mathematical optimization problem. That is, the predistortion fil-

ter’s characterizing parameters are interpreted as a set of variables h to be optimized

and a new frequency-domain measure of transmitter output nonlinearity, referred to

as Weighted Adjacent Channel Power (WACP), is interpreted as a dependent objec-

tive function of those variables B(h). The goal is to find the optimal predistortion

filter parameters which minimize the frequency-domain measure of transmitter out-

put nonlinearity and therefore linearize the transmitter cascade. This optimization

model assumes a nonconvex objective function and therefore utilizes both global and

local optimization algorithms to achieve true global convergence.

In the process of developing these primary wideband novelties, four additional

supporting novelties are developed. These relate to:

• the definition of the Weighted Adjacent Channel Power (WACP) optimization

objective (as introduced above). In addition to conveying complete adjacent

channel spectral regrowth behavior, this objective is able to discriminate be-

tween spectral distortion components and hence control the location of spectral

distortion reduction.

• the demonstration of a new experimental procedure for estimating predistor-

tion filter memory. This procedure is based on sweep-probing the transmitter

with memory specific distortion created by the predistortion filter, and looking

for changes in the signature of the output adjacent channel power spectrum.


• the derivation of predistortion filter parameter optimization schedules, both

Initial Setting and On-Air Adaption, based on the concept of influential sub-

sets. This concept ensures optimization is always performed over the mini-

mally sized variable vector which guarantees complete observability and hence

optimization convergence reliability is always maximal.

• the development of the Distortion Array ; a graphical organizing tool for keep-

ing track of nonlinear distortion components generated by the predistorter-

amplifier cascade. Using this tool, all facets of the predistortion process can

be described both visually and intuitively in the time-domain.

The original contributions of this work are summarized in two full length journal

articles of the IEEE Transactions on Broadcasting and further authenticated by the

application of an international patent in 2011 and successful international search

report in 2012. These research outputs are discussed further in Chapter 3.

The contents of this thesis are spread across 15 chapters. In Chapters 1–3, the

scope of research is set with an introduction to the transmitter linearization prob-

lem, a review of the literature and a specifically defined statement of research. In

Chapter 4, the laboratory transmitter testbed is presented, highlighting the rele-

vance of real hardware testing with DVB-T, WCDMA and DAB signal modulation.

In Chapters 5–7, underlying predistortion theory is developed in preparation for

Chapters 8–12, where the main body of research takes place. In Chapters 13–14,

the new technique’s performance is baselined and tested, with results demonstrat-

ing superiority over current generation techniques. In the concluding Chapter 15,

research contributions are summarized and the future direction of this research is

discussed.

With research context now established, attention is turned to formally introducing

the engineering problem and developing the proposed wideband digital predistortion

technique.

1.1 The Need For Transmitter Linearization

A conventional radio transmitter architecture for mobile basestation and digital

broadcast applications is presented in Figure 1.2. The transmitter consists of two

main stages; the exciter and power amplifier (PA) [6, 245]. The exciter stage trans-

forms the binary input data stream (information) into an RF communication signal

whilst the power amplifier stage boosts RF signal power to a suitable transmission

level.


signalencoder &modulator

exciter stage

DACs

IQ modulator upconverter

IF bandpassfilter

reconstructionfilters

IF LO RF LO

I

Qmask filter

data in

PA stage

PAmodules

Figure 1.2: Conventional radio transmitter architecture

The exciter stage is implemented as follows. A signal encoder and modulator

generates a discrete-time complex baseband communication signal from the binary

input data stream. This signal is applied to a pair of DACs and lowpass recon-

struction filters to form a continuous-time signal. An IQ modulator then converts

this complex baseband signal to a real IF signal which is subsequently translated

to RF via another mixing stage. Average power of the exciter output is typically

in the order of 10 milliWatts. For 3G basestation applications, the exciter performs

DS-CDMA modulation [130,142] whilst for digital television/radio broadcast appli-

cations, it performs OFDM modulation [40,108].

The power amplifier stage is implemented as a parallel combination of power

amplifier plug-in modules. The total number of modules is dependent on the required

output power. In a similar fashion to the overall stage, each power amplifier module

possesses a parallel subamplifier architecture as shown in Figure 1.3.

PA module

RF in RF out

Figure 1.3: Architecture of power amplifier module


This distributed (as opposed to cascaded) approach to amplification allows the

use of smaller RF power FETs [60, 201] which offer higher gain, wider bandwidth

and better phase linearity [217,218]. Other advantages of the distributed approach

include a soft-failure mode and improved thermal management [237]. For the best

tradeoff between efficiency and linearity, each module’s final output stage operates in

Class-AB, typically push-pull [46, 91]. For mobile basestation applications, average

transmitted power is in the order of Watts [1] generally requiring only a single power

amplifier module whilst for digital television / radio broadcast applications, it is in

the order of KiloWatts [236] requiring several power amplifier modules.

Each power amplifier module, and therefore the entire power amplifier stage, ex-

hibits a nonlinear transfer characteristic due to the RF power FETs and Class-AB

output stage. A consequence of this nonlinear transfer characteristic is signal distor-

tion in terms of spectral regrowth [210,278]. Spectral regrowth is classified as either

Co-Channel Distortion (CCD) or Adjacent Channel Distortion (ACD) depending

on where it appears in the output spectrum. Figure 1.4 presents the typical output

spectrum of a digital television transmitter (DVB-T) prior to mask filtering and

without linearization.

RBW 30 kHz

VBW 300 kHz

SWT 2 s

Mixer -40 dBm

A

1RM

Unit dBm

RF Att 40 dB

Ref Lvl

14 dBm

Ref Lvl

14 dBm

2 MHz/Center 670 MHz Span 20 MHz

9 dB Offset

LD

-80

-70

-60

-50

-40

-30

-20

-10

0

10

-86

14

DVB-T

Lower AdjacentChannel Distortion

Co-ChannelDistortion Spectral Mask

Desired Output Spectrum

Allocated TransmissionChannel

Lower AdjacentChannel

Upper AdjacentChannel

Upper AdjacentChannel Distortion

Figure 1.4: Spectral regrowth


The blue dashed trace represents the desired transmitter output spectrum whilst

the green trace represents the actual transmitter output spectrum with CCD and

ACD marked. Note that in this figure, two sets of OFDM carriers are intentionally

removed from the transmission to allow observation of CCD. If these carriers weren’t

removed, CCD would be obscured by the desired transmission spectrum.

As its name suggests, ACD falls outside the transmitter’s allocated transmission

channel. The transmitter’s bandpass mask filter (refer to Figure 1.2) is tasked with

removing this distortion. However, given its finite roll off and attenuation [180,261],

some ACD is still transmitted. This distortion ultimately has the potential to in-

terfere with other users of the spectrum. In order to control such interference,

government regulatory authorities impose strict spectral emission requirements in

the form of spectral masks [2, 64, 65]. The red trace in Figure 1.4 represents the

DVB-T mask for the specific scenario of a co-located digital television transmitter.

The transmitter’s output spectrum must remain within this mask whilst broadcast-

ing otherwise it is deemed noncompliant and penalties apply. ACD is characterized

in the frequency-domain by the Adjacent Channel Power Ratio (ACPR) [5].

In contrast to ACD, CCD falls within the transmitter’s allocated transmission

channel. While a receiver can filter out ACD from the intended transmitter (greater

filter selectivity at IF [308]), it cannot filter out CCD. As a result, CCD interferes

with the intended transmission, resulting in symbol constellation warping / spreading

and therefore symbol detection errors and increased Bit Error Rate (BER) [168].

CCD is characterized by the Error Vector Magnitude (EVM) or Modulation Error

Ratio (MER) in the symbol constellation-domain [66]. As alluded to previously,

CCD can only be observed in the frequency-domain if a portion of the intended

transmission spectrum is removed. For OFDM this means removing carriers whilst

for DS-CDMA this means notch filtering. If this is possible, CCD can be character-

ized in the frequency-domain by the Co-Channel Power Ratio (CCPR) [210].

Modern digital transmission systems utilize OFDM and DS-CDMA signals be-

cause of their superior spectral efficiency and multiple access properties respec-

tively [70]. Examples of such systems include digital television (DVB-T), digital

radio (DAB) and 3G mobile (UMTS WCDMA). OFDM signals are constructed by

adding together many individual carriers (orthogonally frequency spaced) each with

a randomly encoded amplitude and phase [100,161] whilst multi-user DS-CDMA sig-

nals are constructed by synchronously adding together many chipped RRC pulses,

once again, each with a randomly encoded amplitude and phase [77, 271]. This ac-

tion of adding together many individual signal components each with a randomly

encoded amplitude and phase leads to an overall signal with nonconstant envelope

and very high Crest Factor, also known as Peak-to-Average Power Ratio (PAPR).

Table 1.1 presents a comparison of Crest Factors for various signal modulations.


Signal Modulation Crest Factor (dB) Assumptions

FM 0 Theoretical

QPSK 6.2 Random bit stream

16 QAM 8.0 Random bit stream

DAB, DVB-T, WCDMA ≈ 11 Statistically Dependent

Table 1.1: Comparison of Crest Factors for various signal modulations [234]

A clear downside to using these OFDM/CDMA signals is thus the need to

accommodate their very high Crest Factors during physical transmission. Compared

to other communication signals transmitted with the same average power, these high

Crest Factor signals generate greater transmitter distortion (ACD and CCD). This

is because their larger signal peaks drive the transmitter power amplifier into regions

of greater nonlinearity.

One obvious solution to avoid this increased level of distortion is to back off input

signal power to the transmitter power amplifier stage such that each of its modules

is always operating in a linear region. This solution, also known as Output Back Off

(OBO) [204], is unfavorable however as it significantly reduces the efficiency of each

module and necessitates the use of additional parallel modules in order to achieve

the required transmission power. This inefficient utilization of the power amplifier

ultimately leads to increased operational / acquisition costs, cooling requirements

and system footprint. When used, 6–9 dB back off is typical [37].

A more favorable solution, one that avoids OBO, is to incorporate additional

signal processing into the transmitter to further linearize the transfer characteristic

of the power amplifier stage. Many such linearization techniques exist including

Cartesian Loop [211], Envelope Elimination and Restoration (EE&R) [124], feed-

forward [26,251], negative feedback [27], Linear Amplification Using Nonlinear Com-

ponents (LINC) [42], RF predistortion [135] and digital predistortion [198,238]. As

a measure of effectiveness of these linearization techniques, WCDMA power ampli-

fiers are estimated to achieve efficiencies of 3-5% with OBO, 6-8% with feedforward

linearization, 8-10% with current digital predistortion and greater than 15% (pro-

jected) with future digital predistortion [136,186].


1.2 Digital Predistortion Linearization

For high Crest Factor OFDM and DS-CDMA applications, digital predistortion is

both the favored and most promising linearization technique given its cost effective-

ness, superior signal processing capability, potential to adapt and ability to linearize

the entire transmitter; not just the power amplifier stage. The digital predistor-

tion process involves inserting a nonlinear digital filter directly at the output of the

transmitter’s signal encoder and modulator as shown in Figure 1.5.

signalencoder &modulator

exciter stage

DACs

IQ modulator upconverter

IF bandpassfilter

reconstructionfilters

IF LO RF LO

I

Qmask filter

data in

PA stage

PAmodules

nonlineardigital

predistortionfilter

I

Q

Figure 1.5: Placement of digital predistortion filter

The filter’s transfer characteristic is designed to be the inverse nonlinear transfer

characteristic of all downstream transmitter components, but specifically the power

amplifier stage, thereby creating an overall linear transmission cascade.

Theoretically, digital predistortion can allow a power amplifier to be operated at

a higher power with the same level of distortion or at the same power with a lower

level of distortion thus allowing a lower order output mask filter to be used to ensure

the regulatory mask is met [9]. Generally speaking, the latter scenario is pursued.

1.3 Conclusions Reached

Based on this introductory discussion, the following conclusion is reached:

There is a greater requirement for transmitter linearization in OFDM/CDMA

systems due to the nonconstant envelope and high Crest Factor of such

signals, and digital predistortion is the most promising technique to suc-

cessfully perform this linearization requirement.

Since digital predistortion is the way forward, we next perform a literature review

of this technique with the aim of identifying its current state-of-the-art and hence

opportunities for further beneficial research.

Chapter 2

Literature Review of Digital

Predistortion

Digital predistortion was first reported by Grabowski and Davis [89] in 1982. Over

the course of three decades, digital communication systems have evolved in terms of

signal modulation, signal bandwidth, multiple access strategies and power amplifier

hardware [101, 106, 170, 171]. It is this continual evolution of the digital communi-

cation system which drives the flow on evolution of digital predistortion and which

makes such research still relevant today.

The design and analysis of a digital predistortion system is framed by two fun-

damental questions:

1. What architecture will the predistortion filter take?

2. How will the characterizing parameters of this filter be estimated?

Before reviewing the literature guided by these two questions, we first introduce the

concept of power amplifier memory since this memory has a significant influence on

design outcomes.

Concept of Power Amplifier Memory

Ideally, a power amplifier’s transfer characteristic is an instantaneous linear gain.

In practice however, the power amplifier’s output signal is, to some extent, a linear

and nonlinear function of both past and present inputs. This undesirable dynamic

behavior is referred to as amplifier memory. In terms of digital predistortion, it is

an amplifier’s nonlinear memory components which are of main focus and from this

point forward our use of the term memory will refer specifically to these nonlinear

memory components.

Amplifier memory can be categorized as either electrical or electrothermal. Elec-

trical memory is caused by variation in matching and bias network impedances over

9

CHAPTER 2. LITERATURE REVIEW OF DIGITAL PREDISTORTION 10

the input signal bandwidth (frequency response), filter group delays, semiconductor

charge-carrier trapping effects and transistor nonlinear capacitance while electrother-

mal memory is attributed to transistor junction temperature couplings (dynamic self

heating) [28,138,278,279]. Electrical memory components can be both short or long-

term whilst electrothermal memory is generally long-term only [29,48,166,178,228].

The extent of both types of memory is ultimately dependent on the specific

design and manufacture of the amplifier as well as modulated signal characteris-

tics such as Probability Density Function (PDF), Complementary Cumulative Dis-

tribution Function (CCDF), Crest Factor, bandwidth, average power and carrier

frequency [29,105,219,280].

Narrowband Memory

In narrowband applications, where the input signal bandwidth is small compared

to the inverse of the amplifier’s memory duration, the amplifier is said to be quasi-

memoryless. This follows from the fact that if we let x(t) be the input temporal

signal and τ be the amplifier memory duration, then x(t − τ) ≈ Kx(t) where K is

a complex coefficient and hence all memory terms within the amplifier’s nonlinear

input-output functional can be packaged into an equivalent accumulated instanta-

neous effect.

Quasi-memoryless amplifiers are characterized by the Amplitude Modulation –

Amplitude Modulation (AM-AM) and Amplitude Modulation – Phase Modulation

(AM-PM) responses. The AM-AM and AM-PM responses of an amplifier are by

definition the output signal amplitude and output signal phase measured as a function

of input signal amplitude, in both cases with sinusoidal excitation [146,222]. These

responses capture the amplitude and phase distortion of the amplifier, resulting from

nonlinear narrowband memory.

Wideband Memory

In wideband applications, where the temporal variation of the input signal envelope

occurs well within the amplifier’s memory duration, the previous approximation

does not hold and the amplifier must be considered purely dynamic, in which case

AM-AM and AM-PM characterizations of the amplifier are no longer appropriate.

In fact, scattering (hysteresis) will occur in any AM-AM and AM-PM response as

demonstrated in [80,139].

In these wideband scenarios, power amplifier memory manifests itself as fre-

quency dependent behavior, specifically amplitude and phase asymmetries between

the lower and upper spectral regrowth sidebands. This asymmetry is dependent on

both carrier frequency and envelope bandwidth [29,147,155].


The simplest laboratory based method for characterizing these asymmetric mem-

ory effects is to use a two-tone signal. The two-tones, with varied tone spacing,

are applied to the amplifier and the distortion components are measured. If the

distortion components are identical in amplitude and phase regardless of the tone

spacings, then the amplifier is considered memoryless otherwise it is considered dy-

namic [34,146,166]. This process is demonstrated in [34,139,188]. Two-tone testing

is merely a simplification test case of the process of applying a multicarrier or wide-

band digitally modulated signal to the amplifier. In either case, higher spurious

emission generally occurs in the lower frequency sideband [139].

Wideband memory effects are also known as dynamic distortions within the liter-

ature and are one of the limiting factors for distortion cancellation in power amplifier

predistortion [80].

Building on this introduction to power amplifier memory, we now proceed to review

the literature guided by the two fundamental questions on Page 9. As will become

evident, the outcome of both questions is strongly influenced by the communication

system bandwidth in question. That is, the overall predistorter design is different for

narrowband compared to wideband applications.

2.1 Predistortion Filter Architectures

Early Narrowband Applications

The majority of early work in digital predistortion is associated with microwave

high power terrestrial link and satellite / earth station transmitters generally com-

posed of TWT or GaAs FET amplifiers with high order linear signal modulation,

typically 16 - 256 QAM [71,90,127,131,216,254]. Compared to analog FM or digital

PSK/MSK/CPM, this signal modulation has higher spectral efficiency but a non-

constant envelope which inevitably leads to nonlinear amplifier distortion. Given

the narrow percentage bandwidth of these high frequency systems, transmitters are

considered quasi-memoryless and as a consequence the complementary predistorter

nonlinearity is also treated as a memoryless system [216,222]. These instantaneous

predistorters fall into two general categories, data and signal predistorters, depend-

ing on whether the predistorter is placed before or after the pulse shaping filter

respectively [49].


2.1.1 Data Predistortion (Narrowband)

Data predistortion [215,242], also known as prerotation, involves modifying the shape

of the input symbol constellation such that after passing through the nonlinear power

amplifier stage, the output constellation is the desired original shape. It naturally

follows that data predistortion is highly restricted by pulse shaping format and

has constant bandwidth occupancy. If raised cosine (Nyquist) pulse shaping is uti-

lized, data predistortion has the ability to correct constellation warping but has

no effect on spectral regrowth. In contrast, if root raised cosine pulse shaping is

utilized in order to establish matched filtering across the communications channel,

data predistortion is unsuitable due to the pulses being nonzero at adjacent symbol

periods. For the same reason, memoryless transmitter nonlinearity coupled with

root raised cosine pulse shaping leads to intersymbol interference and hence con-

stellation spreading. More advanced versions of the data predistortion technique

strive for better performance by incorporating memory [129, 150, 214] or interpola-

tion [128]. In these respective techniques, constellation predistortion is dependent

on preceding / succeeding symbols and is performed several times within a symbol

interval.

2.1.2 Signal Predistortion (Narrowband)

Signal predistortion acts on the full input signal modulation, not just its constel-

lation, as is the case with data predistortion. Signal predistorters can be further

categorized as either mapping or complex gain predistorters [67,69,174]:

The mapping predistorter [182, 189] indexes its IQ input and maps each such

index to a corresponding IQ output. The two dimensional mapping space is derived

via comparison of the baseband input and demodulated feedback and ultimately

stored in a two dimensional (IQ) Random Access Memory (RAM) Look Up Table

(LUT). In effect, this predistorter directly maps the baseband complex plane to the

RF complex plane and is therefore capable of correcting both the envelope distortion

of the amplifier (AM-AM/AM-PM) as well as the cartesian nonidealities associated

with the modulation process which include carrier leakage, mixer distortion and

mismatches in the analog IQ channels [33, 68, 69, 270]. The disadvantages of this

approach are the excessive number of indices or subsequent trade-off interpolation

required as well as slow convergence rate. [32] provides an analysis of the errors

associated with this input quantization.

The complex gain predistorter [32, 86, 290] is concerned solely with linearizing

the power amplifier, not the entire transmitter, as is the case with mapping predis-


tortion. As a consequence, this predistorter only indexes its input signal complex

envelope (one dimension rather than two) and applies a complex gain correction

based on the amplifier’s AM-AM/AM-PM characteristic. This in effect gives the

transmitter a constant power gain with respect to input signal power. Since index-

ing is one dimensional, the size of the associated RAM LUT is considerably smaller

and interpolation is simplified [269] however conversion to / from rectangular / polar

coordinates as well as complex gain implementation requires additional processing.

As one would expect, since cartesian modulated errors are not corrected for, poorer

performance generally results for complex gain predistorters [174]. Adaptive ver-

sions of this technique are also proposed [119,152] along with statistically weighted

variants [212]. Direct computation of the complex gain correction via cubic spline

interpolation has also been proposed as an alternative to LUT indexing [3].

Analog Predistortion (Narrowband)

It is worth noting that analog IF predistortion [110,191,192] and analog RF predis-

tortion [47, 137, 273] are also proposed for early narrowband applications. However

in addition to being more complex, expensive and dependent on final frequency, the

performance of these predistorters is generally poorer than their digital counterparts

due to their hardware inflexibility and restrictive 3rd order transfer characteristics

based on active devices or Schottky and Varactor diodes [25, 208]. For FET based

power amplifiers, Class AB configurations have significant kinks in their transfer

characteristic which are not well modeled by a cube law [31]. Similarly, 3rd order

distortion components generated by individual stage transistors combine and inter-

modulate with each other at the output stage to create higher order components,

hence requiring higher order predistorters [201].

Later Wideband Applications

The majority of later digital predistortion research is associated with digital cellular

and broadcast transmitters. This is a direct consequence of the enormous growth

and evolution of digital communication systems over the last decade, particularly

into the consumer marketplace [250, 259]. LDMOS FET amplifiers are of prime

focus along with WCDMA and OFDM signal modulation [126, 212], due to the

widespread and current use of the UMTS and DVB/DAB standards. Compared

to narrowband microwave, these more recent applications have significantly lower

carrier frequencies (L-Band and UHF respectively) and therefore larger percentage

bandwidths, making them wideband in nature. Memory effects are therefore ex-

hibited by transmitter power amplifiers (as discussed previously) and hence more


complex predistorters possessing memory are now required as the memoryless com-

pensation techniques already developed prove ineffective [28, 99, 138]. [85] demon-

strates with spectral screenshots the effect that insufficient predistorter memory has

on the transmitter output spectrum, namely reduced inband linearization perfor-

mance and asymmetric lower / upper adjacent channel power. Compared to pure

linear signal modulation utilized by earlier microwave transmission applications,

these new modulation formats exhibit higher signal Crest Factors and hence pose

an even greater requirement for digital predistortion to balance the power amplifier

efficiency / linearity trade off [29,183,226].

Predistortion filter architectures proposed for wideband applications possess

some form of memory and are behavioral data filters [113] rather than physical

or equivalent circuit representations [293]. This trend towards behavioral modeling

is driven by the lower complexity and processing requirements hence higher achiev-

able data throughput and ease of repetitive simulations [117,209,257]. The accepted

choice of behavioral architecture is one that, at minimum, accurately characterizes

the power amplifier and hence has the capacity and potential to generate an equiva-

lent canceling characteristic [203]. Proposed behavioral architectures can generally

be categorized as either advanced or conventional [153]:

Advanced architectures are based on neural networks [24,162,282,294] or fuzzy

methods [153, 158]. Due to their poor analytic nature and slow convergence, these

architectures receive limited coverage in the literature and are nonexistent in com-

mercial implementation.

Conventional architectures are based on nonlinear polynomials or power se-

ries [121, 144]. They include the Volterra Series, Memory Polynomial, Nonlinear

Auto-Regressive Moving Average (NARMA) filter, Hammerstein /Wiener filters,

Twin Nonlinear Two-Box (TNTB) model, as well as variants and hybrids of all [80,

274,289]. These architectures dominate the literature and are discussed below.

2.1.3 Volterra Series (Wideband)

The discrete-time, causal, complex baseband Volterra Series model with maximum

nonlinearity P (odd) and memory M is given by:

y[n] = x[n] +(P−1

2)

∑a=1

⎡⎢⎢⎢⎢⎣

M

∑k1=0

⋯M

∑k2a+1=0

⎛⎝h2a+1[k1, . . . , k2a+1]

a+1∏i=1

x[n − ki]2a+1∏

j=a+2x∗[n − kj]

⎞⎠

⎤⎥⎥⎥⎥⎦(2.1)

In this equation, x[n] and y[n] represent the input and output complex signal en-

velopes respectively and x∗[ ⋅ ] denotes complex conjugation. h2a+1[k1, . . . , k2a+1] is


called the (2a + 1)th order Volterra kernel and the entire set of kernels fully char-

acterizes the model [248]. It is noted that this baseband model only contains odd

order kernels. This is because spectral regrowth created by even order kernels is far

displaced from the allocated transmission channel at RF and subsequently removed

by mask filtering at the output of the transmitter. It is also noted that the Volterra

kernels are complex, containing real and imaginary parts. Being a sum of multidi-

mensional convolutions, this baseband Volterra model is the most general and com-

prehensive behavioral model for dynamic predistortion systems with all other behav-

ioral models, including neural networks and fuzzy methods, being specific cases [289].

The drawback to such a powerful model however is its kernel size. The number of

kernel coefficients which need to be estimated increases exponentially with the de-

gree of nonlinearity and memory length [202,300,301,305]. To overcome this draw-

back, the baseband Volterra model is judiciously pruned to retain only those kernel

coefficients that are expected to influence performance [301]. Numerous pruning

strategies are proposed including Dynamic Deviation Reduction [302–304], Physical

Knowledge [305], V-Vector Algebra [30,298,306], Near-Diagonality Restriction [299],

Base-Band Derived Volterra [160], Dynamic Volterra [151,199] and Volterra Behav-

ioral Wideband [45]. This reduction in complexity increases the Volterra model’s

practical use from weak to mild and strongly nonlinear systems. An alternative

technique for reducing kernel size, one that does not involve pruning, has also been

proposed. This involves projecting the Volterra Series onto a set of Laguerre or

Kautz orthonormal basis functions, making the number of estimated parameters in-

dependent of memory length [97,114,115,300]. Spline basis functions have also been

proposed as a more efficient approximation [241] though basis function projections

have seen limited uptake.

2.1.4 Memory Polynomial (Wideband)

The complex baseband Memory Polynomial [22,57,58,138], also known as the Non-

linear Moving Average (NMA) filter [117,184], is given by:

y[n] = x[n] +(P−1

2)

∑a=1

⎡⎢⎢⎢⎢⎣

M

∑k=0

h2a+1[k] ∣x[n − k]∣2ax[n − k]⎤⎥⎥⎥⎥⎦

(2.2)

With the same notation as (2.1), it is the simplest of the conventional power series

models and corresponds to the Volterra Series with diagonal terms only. In this

sense it is yet another pruned Volterra model but common enough to stand on its

own. An orthogonal variant known as the Generalized Memory Polynomial also

exists [186]. Whilst both forms are theoretically equivalent, the difference lies in

higher order numerical stability (matrix inversion operations) with the orthogonal


variant performing better in the presence of quantization noise and finite precision

processing [220,221]. The Envelope Memory Polynomial, another common augmen-

tation of the pure Memory Polynomial, simplifies implementation by replacing IQ

memory terms with magnitude only memory terms. With this trade off towards

simplicity, nested LUT implementation becomes possible [55,99].

2.1.5 NARMA Filter (Wideband)

The Nonlinear Auto-Regressive Moving Average (NARMA) filter extends the con-

cept of the Memory Polynomial by incorporating diagonalized output memory terms

within the filter’s functional power series [85,184]:

y[n] = x[n] +(P−1

2)

∑a=1

⎡⎢⎢⎢⎢⎣

M

∑k=0

h2a+1[k] ∣x[n − k]∣2ax[n − k] +M

∑j=1

b2a+1[j] ∣y[n − j]∣2ay[n − j]⎤⎥⎥⎥⎥⎦

(2.3)

The relationship between the Memory Polynomial and NARMA model of the non-

linear world is hence akin to the relationship between the Finite Impulse Response

(FIR) and Infinite Impulse Response (IIR) filters of the linear world [183]. The

NARMA structure can reduce the number of coefficients needed to represent short

and long term memory effects, though it can also become unstable due to the non-

linear IIR terms [82].

2.1.6 Hammerstein and Wiener Filters (Wideband)

Whilst the previous three architectures attempt to package both dynamic and non-

linear behaviors into a single power series, the Hammerstein and Wiener filters

separate these behaviors into two standalone subfilters; a static nonlinearity and a

dynamic linearity. As presented in Figures 2.1a and 2.1c, the Hammerstein filter

consists of a cascaded static nonlinearity and dynamic linearity whilst the Wiener

filter consists of the same cascade appearing in reverse order [81,92,93]. This decou-

pling of nonlinear and dynamic behavior simplifies the parameter estimation process.

The static nonlinear subfilter can be implemented as an AM-AM/AM-PM based

LUT or instantaneous nonlinear power series whilst the dynamic linear subfilter can

be implemented as an FIR or IIR LTI filter [83]. Since the Hammerstein and Wiener

filters are structural inverses of each other, they generally appear as matching pairs

in predistortion linearization applications. That is, the predistorter will be modeled

as one form with the power amplifier modeled as the other. Both H-W and W-H

sequences have been proposed. The H-W sequence is applicable to TWT transmit-

ters only, where the combined amplifier and preceding pulse shaping / channel filters

are suitably modeled as a Wiener system. The advantage of this sequence is that

it is possible for the Hammerstein predistorter to be an exact inverse of the power


staticnonlinearity

dynamiclinearity

(a) Hammerstein filter

staticnonlinearity

dynamiclinearity

dynamiclinearity

second ordernonlinearity

(b) Augmented Hammerstein filter

dynamiclinearity

staticnonlinearity

(c) Wiener filter

staticnonlinearity

dynamiclinearity

dynamiclinearity

second ordernonlinearity

(d) Augmented Wiener filter

Figure 2.1: Hammerstein, Wiener and Augmented filter architectures

amplifier given its pre-equalizing connection of LTI subfilters [35, 56, 58, 126]. For

all other transmitter cases, the W-H sequence tends to be applied. Despite an ex-

act inverse relationship not existing and parameter estimation reported to be more

difficult, the W-H sequence tends to apply to broader transmitter classes and hence

can compensate amplifier memory effects more effectively all round [281,292].

Extensions of the conventional Hammerstein and Wiener predistorter architec-

tures have also been proposed. These extensions attempt to address additional

bias / harmonic loading electrical memory effects and hence increase performance.

As presented in Figures 2.1b and 2.1d, the Augmented Hammerstein [164,165] and

Augmented Wiener filters [163] add an additional parallel branch to their dynamic

linear filters. This branch consists of another dynamic linearity cascaded with a

second order nonlinearity hence transforming the original subfilter into a mildly

dynamic nonlinearity with even order components. It is claimed that the subse-

quent even order distortion remixes with the carrier frequency, leading to odd order

canceling products within the main channel [163]. Other variants of the conven-

tional Hammerstein and Wiener architectures are also proposed in the literature

though all instances are applied only to power amplifier modeling, not predistortion.

For reference, these architectures include the Extended Hammerstein [116], Parallel

Wiener [145,258], CombinedWiener-Hammerstein [43,186,243] and Parallel-Cascade

filters [155].


staticnonlinearity

low ordermemory

polynomial

(a) Forward TNTB model

staticnonlinearity

low ordermemory

polynomial

(b) Reverse TNTB model

staticnonlinearity

low ordermemory

polynomial

(c) Parallel TNTB model

Figure 2.2: Twin Nonlinear Two-Box (TNTB) model architectures

2.1.7 TNTB Model (Wideband)

The final wideband predistortion filter architecture is the Twin Nonlinear Two-Box

(TNTB) model. Similar to the conventional Hammerstein /Wiener filters, the ar-

chitecture consists of two subfilters. The difference however lies in replacing the

conventional dynamic linearity with a low order nonlinear Memory Polynomial, in

effect extending the idea of the Augmented Hammerstein /Wiener variants [80].

Three classes of TNTB models exist as presented in Figures 2.2a, 2.2b and 2.2c. In

the forward, reverse and parallel models, the static nonlinearity is placed upstream,

downstream and in parallel to the Memory Polynomial respectively. This model is

reported to out-perform the conventional stand-alone Memory Polynomial architec-

ture whilst reducing the number of model parameters by up to 50% [98].

The previous sections have reviewed both narrowband and wideband predistortion

filter architectures (Question 1, Page 9). Our review now turns attention to predis-

tortion filter parameter estimation techniques (Question 2, Page 9).

2.2 Predistortion Filter Parameter Estimation

Two general strategies exist for estimating predistortion filter parameters; these

being Model Based Derivation (MBD) and Self Learning. The two strategies differ

fundamentally in how they acquire and utilize knowledge of the power amplifier’s

nonlinearity. That is, the MBD strategy relies on power amplifier modeling and

subsequent mathematical inversion to form its predistorter estimate whereas the


Self Learning strategy relies on power amplifier feedback and subsequent predistorter

self tuning. Depending on the form of the feedback, the Self Learning strategy can

be further refined as either Indirect Learning, Direct Learning and Spectral Power

Feedback Learning. These strategies are discussed below.

2.2.1 Model Based Derivation (Narrowband /Wideband)

In its most general form, an MBD strategy is a two step process as presented in

Figure 2.3. The first step consists of modeling the power amplifier using system

identification techniques. The second step consists of deriving the predistortion

filter from the power amplifier model.

For memoryless power amplifiers, both steps are routine with all of the narrow-

band microwave techniques discussed earlier (Data, Mapping and Complex Gain)

being successful examples of this strategy.

The same cannot be said however for wideband applications where power ampli-

fiers exhibit memory effects. In addition to being a more complex system identifica-

tion problem, derivation of the predistortion filter from the power amplifier model

is significantly more involved [138, 303]. For this reason, the MBD strategy is not

generally as common in wideband applications where memory effects must be taken

into account.

It is noted thatMBD has also been referred to as Direct Learning in specific parts

of the literature [206]. This creates potential confusion since one of the prominent

Self Learning strategies (to be discussed soon) is also called Direct Learning. To be

consistent with the majority of literature, we do not mix the terms MBD and Direct

Learning.

The first step of MBD (modeling the power amplifier using system identification

techniques) involves choosing a nonlinear behavioral model to adequately represent

signalmodulator

digitalpredistortion

filter


filters


upconverter

poweramplifier

ADCsfrequency

downconverter,IQ demodulator

tx out

power amplifiermodel

delaycompensation

Figure 2.3: Model Based Derivation (MBD) strategy


the power amplifier, relinearizing this model’s input-output equation and then es-

timating the resulting linear parameters via vectorized linear regression techniques

based on power amplifier input and output signals [160]. Such techniques include

Least Mean Squares (LMS), Recursive Least Squares (RLS) and Fast-Kalman filter-

ing [8,23,38,94,169,173,176]. Relinearization specifically involves creating vectorally

separate signal streams composed of the original signal raised to the appropriate

power. It must be understood that this can only be performed if filtering is linear

with respect to kernel elements, despite the filter output being nonlinearly combined

with the input. This is true for all behavioral models presented previously and is

hence assumed in the following review.

This first step of MBD was pioneered by the power amplifier modeling commu-

nity for communication system simulation but later recognized as being applicable

to the digital predistortion community’s MBD strategy. It is worth noting that

Polyspectral techniques have also been proposed for modeling dynamic nonlinear

systems [255–257] however specific application to MBD predistorter estimation is

still in its inception.

The second step of MBD (deriving the predistortion filter from the power am-

plifier model) is essentially a mathematical inversion problem. Several techniques

have been proposed for this, including P th Order Inverses, Fixed-Point Iteration and

Model Rearrangement.

The P th Order Inverses technique is by far the most common [25, 190, 297,

303]. Pioneered by Schetzen [247, 248], the technique requires the power amplifier

to be modeled as a Volterra system in the first step of MBD. By definition, the P th

order inverse predistorter is then analytically computed as another Volterra system

which removes nonlinear transmitter distortion up to the P th order. Theoretically,

nonlinear distortion greater than P th order remains which must be factored into

the design process. Despite being mathematically rigorous, a major disadvantage of

this technique is the computational complexity involved in both the Volterra system

identification process and also the analytical inverse computation [272], leading to

non-real-time operation. To alleviate this and achieve real-time performance, the

Volterra model must be aggressively pruned in kernel degrees-of-freedom, memory

and nonlinear order which inevitably leads to reduced performance.

The Fixed-Point Iteration technique requires the power amplifier to be mod-

eled as a nonlinear AM-AM/AM-PM characteristic in the first step of MBD. The

mathematical inverse of this model is then computed via standard numerical Fixed-

Point Iteration [132,223]. Mathematically speaking, the uniqueness of the numerical

solution is validated by the Contraction Mapping Theorem [167,207]. Despite being


proposed for OFDM and CDMA applications [112, 140, 291], the wideband perfor-

mance of this technique is limited by the memory constraints of its power amplifier

model.

The Model Rearrangement technique requires the power amplifier to be mod-

eled as a simple dynamic polynomial whose form specifically allows model input

to be rearranged in terms of past model inputs and model output. Following sub-

stitution of the model output with the conditioned transmitter input (this being

the ideal transmitter output after linearization), an expression for the dynamic pre-

distorter input-output equation remains since the model input is merely the pre-

distorter output. This expression can then be solved via vector linear regression

techniques [85, 183, 184] or iteration [138]. Despite adaption being possible, the

downfall of Model Rearrangement is the primitive dynamic polynomial model re-

quired to allow rearrangement to occur. This leads to questionable model fidelity

and is a perfect example of how the MBD strategy is not well suited to wideband

applications.

2.2.2 Indirect Learning (Wideband)

The Indirect Learning strategy was first proposed by Eun and Powers [63]. Its

architecture is presented in Figure 2.4. The term indirect follows from the pre-

distortion filter being derived indirectly from a postdistortion filter. Referring to

Figure 2.4, the power amplifier output is attenuated, downconverted / demodulated

and fed to a nonlinear filter (postdistortion filter). This filter’s architecture is iden-

tical to the proposed predistortion filter. Following a vectorized mathematical re-

linearization of the postdistortion filter’s input-output relationship, its parameters

are estimated via linear regression to minimize the transmitter’s output distortion.

Once converged, the postdistortion filter’s parameters are copied to the predistor-

tion filter [62,79,156,157]. Based on its architecture, this strategy is also known as

signalmodulator

digital predistortion filter(postdistortion filter copy)


filters


upconverter

poweramplifier

ADCs gaincompensation

frequencydownconverter,IQ demodulator

tx out

nonlinearpostdistortion

filter

delaycompensation

Figure 2.4: Indirect Learning strategy


Postdistortion and Translation in the literature.

Whilst intuitive and appealing, the problem with this strategy is its assumption

of nonlinear system commutation, that is, the equivalence of postdistortion and pre-

distortion. Whilst the commutative property may hold for cascaded linear systems,

it doesn’t generally hold for cascaded nonlinear systems and hence the translation

from postdistortion to predistortion can only be considered an approximation with-

out guarantee of adequate performance [81, 184, 206, 295]. [186] points out specific

cases to the contrary [248] however these cannot be relied upon in the general sense.

Another problem with the standard Indirect Learning strategy is that it’s not

capable of seamless adaptive predistortion. That is, predistortion filter adaptation

can only be performed by turning off the current predistortion filter so as to allow

fresh transmitter output data to be collected for postdistortion linear regression.

This in turn means repetitive periods of unlinearized transmission (high distortion)

whilst the predistortion filter is turned off [80]. To overcome this downfall, [59,187]

incorporate a power amplifier model. This model is continuously updated via a

linear system identification technique and allows the postdistorter to be estimated

and translated continuously thereby achieving seamless adaption. The questionable

aspect of this however is the accuracy of the power amplifier model and its flow

on effect in estimating the postdistortion filter. Another method to achieve adap-

tion is proposed by [177] whereby the output of the predistortion filter (as opposed

to input) is used in the optimal tuning of the postdistortion filter. This allows the

postdistortion filter to be estimated and continuously translated to the predistortion

filter whilst on-air. The downside to this approach however is the increased com-

plexity in loop delay compensation (frequency dependent) given the greater signal

bandwidth resulting from predistortion spectral regrowth.

2.2.3 Direct Learning (Wideband)

As its name implies, the Direct Learning strategy is a more direct approach to Self

Learning. Used by transmitter manufacturers such as Rohde & Schwarz, NEC, Eric-

sson and Harris [102,103,194,195,236,237], its architecture is presented in Figure 2.5.

Following a relinearization of the predistortion filter’s input-output equation, vector-

ized linear regression parameter estimation is performed with the objective being to

minimize the mean square error between the baseband attenuated transmitter out-

put and delayed input [22]. This minimization is equivalent to removing transmitter

output distortion.

Compared to the Indirect Learning architecture, the nonlinear power amplifier

is now incorporated into the linear regression optimization process. While this leads

to the distinct advantages of being more direct and avoiding the nonrigorous com-

mutation assumption, it leads to three major problems of its own:


signalmodulator


filter


filters


upconverter

poweramplifier

ADCsgain

compensation

frequencydownconverter,IQ demodulator

tx out

delaycompensation

Figure 2.5: Direct Learning strategy

1. Computation and implementation cost is increased significantly [159]. Linear

regression gradient estimation must be performed either analytically, whereby

a nonlinear model of the power amplifier must first be derived and generally

adapted similar to the first step of MBD [125, 126, 295], or non-analytically

whereby predistorter parameters are incremented in a finite-differences sense

to probe the power amplifier’s response. This probing must be controlled to

prevent unintentional on-air distortion in the adaption process. In the Indirect

Learning strategy, analytical gradient estimation is routinely performed based

on the known postdistortion filter architecture.

2. The inherent assumption of a quadratic error criterion does not hold in the

local linear regression optimization process [22, 78, 177]. In linear regression

theory, the error criterion can only be assumed quadratic if the signal being

subtracted from the desired response (to form the error signal) is the direct

(or linearly filtered) output of the filter being regressed [104, 285]. In the

Indirect Learning and MBD strategies, this is indeed true and hence they can

assume a quadratic criterion. In the Direct Learning strategy however, the

nonlinear amplifier sits between the signal being subtracted from the desired

response and the filter being regressed (predistorter). As a result, the quadratic

assumption does not hold and the proposed local optimization process cannot

guarantee convergence to the global minimum, with concerns remaining about

local inferior convergence.

3. Delay and gain compensation must be implemented within the feedback paths

forming the error criterion as shown in Figure 2.5. While not such a problem

for narrowband systems, this poses a significant problem for wideband systems

where compensation becomes frequency dependent. This point is reinforced

by the fact that signal bandwidth in the output feedback path is at least

nine times that of the original wideband modulated signal given its adjacent


channel distortion components. If compensation is not performed correctly,

the error criterion on which to adapt does not represent the true state of

linearization and the optimization process converges to a biased and hence

performance degrading value [84, 119, 288, 290]. Time delay mismatch from

the input feedback path can also be misinterpreted and considered as memory

effects [80, 172]. Typically, the resolution for delay alignment is lower than

the signal sampling rate. This requires signal up-sampling and down-sampling

during the delay estimation and compensation process [163]. [288] also points

out that with the conventional feedback path, changes in temperature for

example may cause the power amplifier gain to change which then makes the

feedback power level adjustment suboptimal. This means that not only must

one adapt for changes in the power amplifier, one must also follow up by

adapting components in the feedback path, all of which are dependent on each

other, leading to stability issues. While these feedback loop impairments must

be equally managed in the Indirect Learning strategy, errors associated with

the postdistortion translation process appear to attract greater scrutiny. As an

aside, frequency dependent modulator / demodulator and DAC/ADC errors

also degrade predistortion performance where feedback is concerned. While

being more of an early design aspect rather than a learning strategy issue,

measures must be taken to avoid excessive performance degradation [33, 68,

226,253,268,270].

Combined Indirect & Direct Learning

Combining both the Indirect and Direct Learning strategies has also been proposed

in [35, 36]. Here, the predistorter is first estimated Indirectly and then refined

Directly . Despite exhibiting improved convergence properties, this combined learn-

ing strategy has seen limited uptake most likely due to its doubled implementational

cost.

2.2.4 Spectral Power Feedback Learning (Wideband)

To specifically overcome the third pitfall of the Direct Learning strategy, that is time-

domain feedback loop compensation error, Stapleton et al. [107, 262–264] propose

working in the frequency-domain with their Spectral Power Feedback Learning (SPFL)

strategy. Demonstrated in Figure 2.6, the power within a small spectral region of

the transmitter’s adjacent channel is measured and used as the objective function for

a generic single objective optimization of the predistortion filter parameters [75,200].

Since time delay and power level compensation is avoided, a far more robust, reliable

and simple measure of transmitter output nonlinearity results.


signalmodulator


filter


filters


upconverter

poweramplifier

spectralpower

measurement

objectivefunction

formulation

tx out


Figure 2.6: Spectral Power Feedback Learning (SPFL) strategy

The key point to understand here is that theDirect Learning strategy implements

time-domain signal feedback whereas this strategy implements frequency-domain

information feedback ; the difference being the elimination of time delay and power

level compensation (and associated error) in the feedback path. Spectral power is

derived by bandpass filtering and power detection although spectral convolution has

also been proposed [265].

The SPFL strategy should not be confused with implementations which use

ACP measurements for distortion monitoring and adaption control purposes but

then revert back to the time-domain Direct Learning strategy to perform actual

parameter estimation [112,120,194]. SPFL uses spectral power measurement as the

objective in the parameter estimation process.

Another advantage of SPFL, being based on generic nonlinear optimization, is

that it inherently assumes a nonconvex objective function and hence can potentially

use both global and local optimizers to achieve convergence. This eliminates the

second pitfall of the Direct Learning strategy.

2.3 Opportunity For Further Beneficial Research

Figure 2.7 summarizes the findings of this literature review in terms of system band-

width, predistortion filter architecture and predistortion filter parameter estimation

strategy. Section references are also included to aid navigation within the review.

From a research perspective, the aim of this review is to identify the current

state-of-the-art of digital predistortion and hence identify opportunities for further

beneficial research. We identify the SPFL strategy as the basis of such beneficial

research for the following reasons:

1. Since the strategy implements frequency-domain information feedback, and is

hence not limited by compensation error associated with time-domain signal


feedback, it appears more suited to current and future wideband applications

than does the Direct and Indirect Learning strategies. It can only be expected

that the continual increase in signal bandwidth will further expose the feedback

weaknesses of the Direct and Indirect Learning strategies.

2. Stapleton et al. [107,262–264] have only presented the strategy in terms of sys-

tems with linear signal modulation (QAM/QPSK). To the best of our knowl-

edge, no further investigations have been performed for wider-band, higher

Crest Factor applications such as CDMA and OFDM.

Based on the above reasoning, we specifically endeavour to investigate the SPFL

strategy’s promising potential to linearize current and future wideband communica-

tion systems.

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§2.1.1

§2.1.2

§2.1.3

§2.1.4

§2.1.5

§2.1.6

§2.1.7

§2.2.1

§2.2.2

§2.2.4

§2.2.3

§2.1

§2.2

Figure

2.7:

Summaryof

literature

review

Chapter 3

Research Scope & Outputs

In this chapter, the scope of the research endeavor will be discussed. In this context,

the scope will include the statement of research, a list of target applications, all

assumptions and finally the delimitations [154]. After reading this chapter, one will

have a clearer understanding of the boundaries of this research.

3.1 Statement of Research

This research will demonstrate adaptive digital predistortion for wideband high

Crest Factor applications based on the specific concept of Spectral Power Feedback

Learning. This encompasses both aspects of predistortion filter architecture and

parameter estimation.

3.2 Target Applications

In terms of the above research statement, wideband high Crest Factor applications

specifically refers to the following medium to high power (Watt –KiloWatt) radio

transmitter standards:

• 3G UMTS WCDMA mobile basestation downlink

– Modulation Type: DS-CDMA

– Modulation BW: 4.096MHz

– Channel BW: 5MHz

– Frequency Band: UHF 800 – 915MHz

• DVB-T digital terrestrial television broadcasting

– Modulation Type: OFDM


28

CHAPTER 3. RESEARCH SCOPE & OUTPUTS 29

– Channel BW: 7MHz

– Frequency Band: UHF 520 – 820MHz (Broadcast Bands IV/V)

• DAB digital radio broadcasting (Transmission Mode I)

– Modulation Type: OFDM


– Channel BW: 1.75MHz

– Frequency Band: VHF 174 – 230MHz (Broadcast Band III)

All analysis and performance testing carried out in later chapters will thus be

presented in the context of these target applications. For example, a Class-AB

push-pull power amplifier is chosen as part of the laboratory transmitter testbed

(discussed in Chapter 4) specifically to replicate the distortion characteristics of

these transmitters during algorithm testing.

It is also noted that the specific transmission frequencies and bandwidths listed

above are based on the Australian Communications & Media Authority (ACMA)

RF spectrum plan [13,14] and licensed provider allocation within Australia [7, 15].

3.3 Assumptions

Assumptions Relating To The Reader of This Thesis

It is assumed that the reader of this thesis has a background in RF electronics,

communications and signal processing. It is also assumed that the reader is familiar

with object oriented programming, particularly the C++ language, and has access to

Microsoft Visual Studio (or an equivalent IDE text editor) for viewing and navigating

software source code. The reader is also encouraged to peruse the accompanying

DVD as it contains material referenced within this thesis.

Assumptions Relating To The Writing of This Thesis

Software source code associated with this research will not be collated in an appendix

of this thesis. Instead, it will be collated electronically on the accompanying DVD

within the top level Software directory and referenced appropriately. The reason

being, source code is written with horizontal span greater than that of A4 portrait,

courtesy of the generous span of the IDE text editor. Blindly copying this code across

to an A4 portrait page ultimately leads to line breaking and therefore fragmentation

of the original formatting. Source code should be viewed electronically, with proper

formatting, using a text editor.


It is also noted that in cases where a lengthy mathematical derivation would

disrupt the flow of ideas at a particular point in the thesis, that mathematical

derivation is forced to Appendix B and only the final result (plus reference to the

Appendix) is inserted within the thesis.

Assumptions Relating To Algorithm Testing

In this research, algorithm testing is performed on a transmitter testbed assem-

bled within the laboratory. As will be discussed in Chapter 4, the components of

this testbed replicate the characteristics of a commercial transmitter and therefore

testing results are highly relevant.

3.4 Delimitations

Delimitations Relating To Measures of Nonlinearity

The OFDM and DS-CDMA signals used in the target applications of Section 3.2

are continuous-spectra signals with substantial bandwidth. For this reason, the

following measures of amplifier nonlinearity, applicable only to single-tone, two-tone

or narrowband signals, will not be discussed:

• Compression

• Intercept Points

• AM-AM/AM-PM

This thesis will use the following, more appropriate measures of amplifier nonlinearity:

• Adjacent Channel Power Ratio (ACPR)

• Co-Channel Power Ratio (CCPR)

Delimitations Relating To Implementation

The focus of this research is on predistortion filter architecture and parameter esti-

mation. Optimal implementation of these components in software / firmware / hardware,

for the purpose of minimizing computation or memory requirements, will not be con-

sidered.

Delimitations Relating To Transmitter Channels

This research only considers single-channel radio transmitters. This is in contrast

to multi-channel radio transmitters which combine multiple exciter outputs, each


with a different channel or carrier frequency, prior to the power amplifier stage.

In addition, frequency agile channels and channel switching will not be considered.

Once a service channel has been allocated, that channel will remain fixed.

3.5 Commercialization

One clearly defined output from this research activity is Intellectual Property and

the potential for commercialization. A Provisional and International patent appli-

cation entitled A method and system for linearising a radio frequency transmitter

was filed on 04 January 2011 and 23 December 2011 respectively. This was followed

by an International Search Report being generated on 12 July 2012. Commercial-

ization opportunities are currently being sought with the help of commercialization

company UniQuest (www.uniquest.com.au). Given its length, the full document set

representing the patent description, drawings, claims, application filings and search

report is forced to the accompanying DVD under the top level Patent folder. The

summarizing Claims section is however included in Appendix C of this thesis.

3.6 Publications

Publications are another clearly defined output of this research activity. Two jour-

nal articles have been published in the IEEE Transactions on Broadcasting. The

first article entitled Adaptive Digital Predistortion for Wideband High Crest Factor

Applications Based on the WACP Optimization Objective: A Conceptual Overview

outlines the concepts of the proposed method of digital predistortion and provides

preliminary results. The second article entitled Adaptive Digital Predistortion for

Wideband High Crest Factor Applications Based on the WACP Optimization Objec-

tive: An Extended Analysis is a more indepth presentation of the proposed method,

discussing implementation aspects and providing further adaption results. Both

publications are included in Appendix D of this thesis.

Chapter 4

Laboratory Transmitter Testbed

A radio transmitter testbed assembled in the laboratory is used to develop and

test all algorithms associated with this research. A block diagram and photo of

this testbed is presented in Figures 4.1 and 4.2 respectively. It is noted that in

both figures, testbed components are labeled uniquely with a red letter in order to

facilitate component matching across figures.

A Rohde & Schwarz AMIQ IQ Modulation Generator (C) implements the digital-

to-analog conversion and reconstruction filtering processes whilst a Rohde & Schwarz

SMIQ Vector Signal Generator (D) implements the IQ modulation and frequency

upconversion processes. Power amplification is performed by a solid state, 25Watt

Class-AB push-pull power amplifier (G). Preceding this power amplifier is a driver

amplifier (E). The output of the power amplifier is subsequently fed to a dummy load

(K). A Rohde & Schwarz FSIQ Spectrum Analyzer (L) measures transmitter output

Power Spectral Density via a directional coupler (J) at the output of the power

amplifier. Signal encoding and modulation (A), predistortion filtering (B), objective

function measurement (M) and mathematical optimization (N) are all implemented

in software running on a standalone PC (O) with GPIB capability. The user is able

to externally configure, control and monitor the testbed via a DOS shell console

interface (P) offering numerous built in menu options. All software is written in

C++ with object-oriented design [123, 267]. This software and all test instruments

communicate via GPIB. A further discussion of these testbed components is given

in the following.

4.1 AMIQ IQ Modulation Generator (C)

Technical specifications of the Rohde & Schwarz AMIQ IQ Modulation Generator

(Model: 1110.2003.04) are given in [229,230]. The instrument consists of two chan-

nels (IQ) of SDRAM, digital-to-analog conversion and reconstruction filtering.

32

sign

al e

ncod

er&

mod

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or(A

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Figure

4.1:

Block

diagram

ofthelaboratorytran

smittertestbed

O

PLC D

F

H

I

GE

KJ

Figure

4.2:

Photoof

thelaboratorytran

smittertestbed

CHAPTER 4. LABORATORY TRANSMITTER TESTBED 35

The complex predistortion filter output signal is transfered to the instrument’s

SDRAM via GPIB and clocked to the DACs at the preset sampling rate. The

AMIQ’s output clocking frequency can be set in the range 10-100MHz while its

reconstruction filters can be set to either a 2.5MHz or 25MHz cutoff frequency.

Knowing that the predistortion filter will generate high order spectral regrowth,

there is no choice but to set the reconstruction filters to the maximum 25MHz cut-

off frequency for each of the target applications. The following sampling frequencies

(within the 100MHz maximum limit) are chosen for each of the target applications:

• DAB: 65.536MHz

• WCDMA: 92.160MHz

• DVB-T: 64MHz

Two points are worth clarifying with respect to the DVB-T signal:

1. The sampling rate of the DVB-T signal is lower than that of the DAB and

WCDMA signals despite exhibiting a greater continuous-time bandwidth. This

is because the next power of two IFFT would bring the sampling rate to greater

than the 100MHz maximum. The current sampling rate is satisfactory however

as it oversamples the DVB-T signal by at least the highest predistortion filter

nonlinearity (9th order as will become evident in Chapter 8) and therefore

avoids spectral regrowth aliasing.

2. Theoretically, the 25MHz reconstruction filters can only fully reconstruct up

to the 7th order components of the predistorted DVB-T signal. In practice

this is not a problem however as the tailing end of the 9th order predistor-

tion spectrum situated beyond the reconstruction filter cutoff frequency (and

therefore linearly distorted) will not have any effect on the final predistortion

process as it will fall beneath the spectrum analyzer noise floor. In conclusion,

the useful portion of the 9th order predistortion spectrum will be adequately

reconstructed.

In this transmitter testbed, a scaling mechanism is built into the software be-

tween the predistortion filter output and AMIQ input, to control and monitor how

much of the DAC’s full scale range is being utilized at any specific time. Prior to

predistortion, this scaling mechanism is set such that peak signal excursion is ap-

proximately 30% of full scale. This is for good reason. During the predistortion

process, signal peaks expand to compensate for the compression effects of the power

amplifier and as a result, peak signal excursion grows from the intial 30% to ap-

proximately 80% of full scale. If this growth in peak signal excursion wasn’t taken

into account in the initial setting of the scaling mechanism, the DACs would end up


nonlinearly distorting and clipping the signal during predistortion, leading to major

spurious components.

As discussed in Chapter 1, a downside to using these OFDM/CDMA signals

is the need to accommodate their large Crest Factors during physical transmission.

This applies not only to the power amplifier but also to the DACs. With the scaling

mechanism discussed above set to accommodate the signal peaks, the average signal

excursion is significantly less than 30% of full scale due to the high Crest Factors in-

volved1. With only a small fraction of the full scale range therefore being utilized on

average, the finite resolution limitations of the DACs become apparent. In general,

broadband quantisation error can be expected, leading to a raised spectral noise

floor and therefore reduced spectral dynamic range. In the AMIQ, DACs possess 14

bits of resolution leading to a spectral dynamic range of approximately 50 dBc.

4.2 SMIQ Vector Signal Generator (D)

Technical specifications of the Rohde & Schwarz SMIQ Vector Signal Generator

(Model: SMIQ06B 1125.5555.06) are given in [232,233]. The instrument is operated

as a standard IQ modulator and RF upconverter (no additional model features

included) with the following non-default settings to ensure greater output linearity

and level repeatability:

• Output Mode - Low Distortion

• Automatic Level Control - Off - Table Mode

Recalibration of the level control table is performed regularly as outlined in the ap-

plication note [19]. The instrument’s output power is controlled by a level setting

which internally configures the output attenuation. The maximum linear output

power of the instrument is not enough to drive the power amplifier into severe non-

linear operation hence the need for a driver amplifier (E) applied at the instrument’s

output. It follows that the instrument’s level setting is used to indirectly control the

input drive level of the power amplifier.

4.3 Driver Amplifier (E)

The driver amplifier used to boost SMIQ output power is the Mini-Circuits Gali-52.

Technical specifications can be found in the data sheet included in Appendix E. This

is a wideband amplifier with approximately 20 dB of gain and a 15.5 dBm output

power rating. The actual output power necessary to drive the power amplifier into

1Expansion of signal peaks to 80% of full scale during predistortion has little effect on the averagepower.


saturation is approximately -3 dBm. This is 18.5 dB below the output power rating

and hence the driver amplifier operates linearly.

4.4 Power Amplifier (G)

The power amplifier used in the laboratory transmitter testbed is the OPHIR RF

5303038. This power amplifier’s characteristics are consistent with those of our tar-

get application power amplifier modules and therefore it can be expected to replicate

their nonlinear behavior in a laboratory test environment. It is a solid state, broad-

band, 25Watt Class-AB push-pull power amplifier. Technical specifications can be

found in the data sheet included in Appendix E. Transmitter power amplifier mod-

ules operate in Class-AB push-pull to obtain the best tradeoff between efficiency

and linearity. In light of reduced manufacturing cost and spares inventory, they are

also broadband in nature, avoiding channelized designs [40,194,237].

Fan forced cooling is applied to the power amplifier (as can be seen in Figure 4.2

on Page 34) with temperature monitored via a thermocouple (I) attached to the

amplifier’s casing. Amplifier temperature is controlled via the fan speed and there-

fore airflow over the casing. A greater airflow reduces operating temperature and

vice versa. This simple but effective means of manipulating amplifier temperature

is used in Section 14.2 to vary the amplifier’s nonlinear transfer characteristic and

therefore to test algorithm adaptivity.

4.5 FSIQ Spectrum Analyzer (L)

Technical specifications of the Rohde & Schwarz FSIQ Spectrum Analyzer (Model:

FSIQ26 1119.6001.27) are given in [231]. It is noted that this instrument is techni-

cally referred to as a signal analyzer as it has two modes of operation; a spectrum

analyzer mode and a vector analyzer mode. The vector analyzer mode allows the

analysis of digital modulations in the constellation-domain, however this mode will

not be used. The instrument is strictly set to spectrum analyzer mode and will

therefore be referred to as such.

The spectrum analyzer is configured such that the input attenuation is coupled

to the reference level. In light of the signal’s initially high and continually growing

Crest Factor (growth occurs during predistortion), the reference level must be set

higher than normal to ensure the input mixer is not overdriven. A further safe guard

against overdriving the input mixer is the non-default setting:

• Attenuation Auto Mode - Low Distortion

which adds an additional 10 dB to the auto coupled input attenuation.


It is worth noting that for high Crest Factor signals, maximum theoretical dy-

namic range is actually achieved when the input attenuation is decoupled from the

reference level (opposite to above) and set such that the peak input signal power is

approximately 10 dB below the 1 dB compression point of the signal path prior to

the IF filter [287]. However in practice, decoupling of the input attenuation is un-

necessary and will not provide any extra performance in our case as it is the AMIQ’s

substantial quantization noise which determines the overall dynamic range.

To obtain stable and repeatable power spectrummeasurements, an RMS detector

is selected [287]. Resolution bandwidth is set to 30KHz (0.45% to 2% of signal band-

width), resulting in adequate frequency resolution and a practical 2 second sweep

time [225]. Video bandwidth is set to 10 times the resolution bandwidth, thereby

avoiding averaging and allowing the true power spectrum to be measured [72].

4.6 Signal Encoding and Modulation (A)

The signal encoding and modulation process is encapsulated in software as an object-

oriented class [249]. Given the slightly different encoding and modulation require-

ments, a separate class is defined for DAB, DVB-T and WCDMA. During software

build, conditional compilation directives (#if, #elif) allow the developer to select

which one of these classes to instantiate as the testbed’s encoder/modulator object.

Class data members include:

• the modulated signal:

– that which is applied to the predistortion filter

– stored as a complex vector

• parameters of the modulated signal:

– number of OFDM carriers (DAB and DVB-T classes)

– guard interval (DAB and DVB-T classes)

– flag to remove carriers and monitor CCD (DAB and DVB-T classes)

– encoding type (DAB, DVB-T and WCDMA classes)

– chipping rate (WCDMA class)

– spreading factor (WCDMA class)

– user channels (WCDMA class)

– RRC rolloff factor (WCDMA class)


Class member functions include:

• generating the modulated signal

• writing the modulated signal to file

• reading the modulated signal from file

• computing modulated signal characteristics such as Crest Factor and peak power

• notch filtering the modulated signal to monitor CCD (WCDMA class only)

The DAB class is configured to encode and modulate according to the DAB digital

radio broadcasting standard (Transmission Mode I) as follows:

• Modulation Type: OFDM

• Modulation BW: 1.537MHz

• Channel BW: 1.75MHz

• Encoding: QPSK

• Symbol Duration: 1ms

• Guard Interval: 246 µs

• Carriers: 1536 spaced 1KHz apart

The DVB-T class is configured to encode and modulate according to the DVB-T

digital terrestrial television broadcasting standard (either 8K or 2K Mode) as follows:

• Modulation Type: OFDM


• Channel BW: 7MHz

• Encoding: 16 or 64 QAM

• Symbol Duration:

– 8K Mode: 1024 µs

– 2K Mode: 256 µs

• Guard Interval: 1/4, 1/8, 1/16 or 1/32 of symbol duration


• Carriers:

– 8K Mode: 6817 spaced 976.5625 Hz apart

– 2K Mode: 1705 spaced 3.90625 KHz apart

TheWCDMA class is configured to encode and modulate according to the 3G UMTS

WCDMA standard as follows:

• Modulation Type: DS-CDMA


• Channel BW: 5MHz

• Encoding: QPSK

• Chipping Rate: 3.84Mchips/s

• RRC Rolloff Factor: 0.22

• User Channels: 512

It is noted that the number of WCDMA user channels above (512) is set higher than

the standard 256. This is done intentionally to increase the signal’s Crest Factor

and hence create a more difficult test scenario [122,227].

A plot of the Complementary Cumulative Distribution Function (CCDF) repre-

sentative of a DVB-T, WCDMA and DAB signal taken over a finite 1 000 000 signal

sample duration is presented in Figure 4.3. Also included for comparison are the

CCDF plots for FM modulation, QPSK and 16 QAM. These CCDF plots are derived

using the envelope power approach [18]. The CCDF plot of a signal represents the

probability (y-axis) of the signal’s instantaneous power exceeding its mean power

by a specified value (x-axis) [235]. The Crest Factor of a signal can subsequently

be derived statistically from its CCDF plot as the x-axis intercept point [234]. The

Crest Factor derived this way clarifies Table 1.1’s Crest Factor values.

4.7 Predistortion Filtering (B),

Objective Function Measurement (M) and

Mathematical Optimization (N)

Since these testbed components are the focus of our research endeavor, their details

and workings will be developed throughout the remainder of this thesis. At this stage

however, it is sufficient to understand that these components are implemented in soft-

ware running on the standalone PC (O) and configured via the console interface (P).


1E-05

1E-04

1E-03

0.01

0.1

*Att 30 dB

AQT 91.14ms

1Sa

View

Ref 20.00 dBm

CF 670.0 MHz Mean Pwr + 20.0 dB

2Sa

View

3Sa

View

4Sa

View

1000000 samples

DVB-T / WCDMA / DAB

QPSK

16 QAM

FM

CF 670.0 MHzP

rob

abili

ty

0 2010 12 14 16 182 4 6 8

Instantaneous Power Exceeding Mean Power (dB)

Figure 4.3: Complementary Cumulative Distribution Functions

4.8 Console Interface (P)

The transmitter testbed is configured and controlled via a console DOS shell user

interface menu. After system initialization, program execution enters a repetition

and selection structure which implements the user interface menu. The available

menu options allow the user to:

• Measure ACP, CCDF and Crest Factor data

• Manipulate spectrum analyzer traces to log past / present ACP reduction

• Set the AMIQ DAC scaling mechanism

• Enable / disable individual predistortion filter nonlinearities

• Configure predistortion filter coefficients (reset / load / save / display / increment)

• Configure optimization objective function characteristics


• Generate / load / save encoded and modulated signals

• Run individual or sets of optimization algorithms in the form of a schedule

• View testbed setup details

The console interface also provides real-time textual and numeric feedback, allowing

the user to monitor the state of the testbed and its predistortion algorithms at all

times. It follows that with this console interface, the researcher is able to config-

ure, control and monitor all the experiments involved in algorithm development and

testing.

With the first four chapters of this thesis devoted to introducing the research en-

deavor and testing environment, we now turn attention towards accomplishing our

specific statement of research.

Chapter 5

Volterra Series Modeling of

Amplifier & Predistorter

In this chapter, we introduce the nonlinear Volterra Series and discuss its inter-

modulating and spectral regrowth properties. With the RF power amplifier then

modeled as such, we convert the RF transmitter model to its baseband equivalent

thereby showing that the resulting predistortion filter takes on a variant architecture

of the pure Volterra Series called the Baseband Volterra Series.

As discussed in the Literature Review of Chapter 2, those predistortion filter archi-

tectures proposed for today’s wideband applications possess some form of memory

and are behavioral models rather than physical circuit level representations. Memory

is required to compensate for the dynamics of the power amplifier (now modulated

with wideband signals) while behavioral models are favored due to their lower com-

plexity and processing requirements. Conventional behavioral models with memory

include the Volterra Series, Memory Polynomial, NARMA filter, Hammerstein and

Wiener filters, TNTB model as well as variants and hybrids of each. The reader is

directed to page 14 for further details of these models.

In this research, the Volterra Series is chosen as the foundation model for the

power amplifier and predistortion filter for the following high level reasons:

Model Generality - Of all models, the Volterra Series is the most general. If the

Volterra Series is not capable of representing a nonlinear system, then no other

model will [280,289].

Mathematical Tractability - Of all models, the Volterra Series is the most math-

ematically tractable. Compared to other models, it provides the greatest in-

sight into nonlinear system interaction and behavior [117,278].

43

CHAPTER 5. VOLTERRA SERIES MODELLING 44

Application to Future Wider Band Systems - Of all models, the Volterra Se-

ries exhibits the greatest degrees of freedom in memory thus offering the great-

est potential for compensating those complex dynamic nonlinearities expected

of future wider band systems [209,274]. It is anticipated that all other models

will become inadequate with growth in bandwidth.

It is understood that any foundation Volterra Series model will have a large kernel

size and some form of pruning will be necessary for final implementation of the pre-

distortion filter. In Chapter 12 we address this pruning aspect. Prior to Chapter 12

however, we specifically ignore pruning in order to maximize insight into algorithm

development. That is, in terms of nonlinear system modeling, we favor the idea of

staying as general as possible for as long as possible, only specializing (in this case

pruning) when needed. This allows our work to develop freely without fear of being

pushed in a specific direction based on the limitations of any specific model. The

benefit of this guiding mindset is that our work will remain applicable to a broader

range of future wideband standards. That is, as will become apparent in later chap-

ters, the pruning strategy will be the only part of the proposed technique needing to

be matched to a varying bandwidth modulation standard. In this sense, the pruning

strategy can be seen to act as the binding link between a constant technique and

the forever changing application space.

5.1 Introduction To The Volterra Series

The Volterra Series was first studied by mathematician Vito Volterra [276, 277].

It is suitable for modeling mildly nonlinear1 time invariant systems and can be

represented in either operator or functional form. In conjunction with Figure 5.1,

the operator form is given by:

y(t) = H[x(t)] =∞∑n=1

Hn[x(t)] (5.1)

where H[ ⋅ ] represents the system Volterra operator, x(t) and y(t) represent systeminput and output respectively and Hn[ ⋅ ] represents the nth order Volterra operator,

that is, the individual nth order nonlinearity.

The continuous-time, causal, pure Volterra Series in functional form is given by:

y(t) =∞∑n=1

⎛⎝

∞

∫0

⋯∞

∫0

hn(τ1, . . . , τn) x(t − τ1)⋯x(t − τn) dτ1⋯dτn⎞⎠

(5.2)

13rd ≫ 5th ≫ 7th order distortion which is consistent with our intended application


H[ ⋅ ]x(t)x(t) y(t)y(t)

H1[ ⋅ ]

Hn[ ⋅ ]

H∞[ ⋅ ]

⋮

⋮

Figure 5.1: Operator form of Volterra Series

where hn(τ1, . . . , τn) represents the nth order Volterra kernel. The entire set of

kernels (n = 1 to ∞) fully characterizes the nonlinear Volterra system. Kernels are

real in general and are functions of input memory. Equating (5.1) and (5.2) gives

the functional form of the nth order Volterra operator:

Hn[x(t)] =∞

∫0

⋯∞

∫0

hn(τ1, . . . , τn) x(t − τ1)⋯x(t − τn) dτ1⋯dτn (5.3)

In our analysis, kernel symmetry is assumed, that is hn(τ1, . . . , τn) = hn(τn, . . . , τ1),without loss of generality since any nonsymmetric kernel can be symmetrized [240,248].

To understand the nonlinear Volterra interaction between input signal components,

let the input x(t) now be represented as a general sum of weighted signal components:

x(t) =A

∑a=1

casa(t) for A ≥ 1 (5.4)

The response of the nth order Volterra operator then becomes:

Hn[A

∑a=1

casa(t)] =A

∑a1=1

⋯A

∑an=1

ca1 ⋯ can

⎛⎝

∞

∫0

⋯∞

∫0

hn(τ1, . . . , τn) sa1(t − τ1)⋯san(t − τn) dτ1⋯dτn⎞⎠

(5.5)


The right hand bracketed term in (5.5) is formally known in the literature as the

n-linear Volterra operator and represented as:

Hn{sa1(t), . . . , san(t)} =∞

∫0

⋯∞

∫0

hn(τ1, . . . , τn) sa1(t − τ1)⋯san(t − τn) dτ1⋯dτn

(5.6)

It is given this name since it is linear in each argument when all others are held fixed.

With this formal representation, the response of the nth order Volterra operator

becomes:

Hn[A

∑a=1

casa(t)] =A

∑a1=1

⋯A

∑an=1

ca1 ⋯ can Hn{sa1(t), . . . , san(t)} (5.7)

From this expression, we can make some very important observations:

1. Unlike a linear system where the response to a sum of weighted signal com-

ponents would be a sum of weighted component responses, we now have in-

teraction between the different input signal components, leading to output

inter-modulation responses.

2. Since multiplication in the time-domain is equivalent to convolution in the

frequency-domain, specifically referring to the product terms within the inte-

gral of the n-linear Volterra operator (5.6), output spectral regrowth is gener-

ated by the inter-modulating signal components.

To investigate this spectral regrowth further, we turn to a frequency-domain repre-

sentation of the Volterra Series. Let the input x(t) now be represented as a general

accumulation of weighted spectral components:

x(t) =∞

∫−∞

X(f) ej2πft df (5.8)

The response of the nth order Volterra Series operator to this input is:

Hn

⎡⎢⎢⎢⎢⎣

∞

∫−∞

X(f) ej2πft df⎤⎥⎥⎥⎥⎦=∞

∫−∞

⋯∞

∫−∞

X(f1)⋯X(fn) ej2π(f1+⋯+fn)t

⎛⎝

∞

∫0

⋯∞

∫0

hn(τ1, . . . , τn) e−j2π(f1τ1+⋯+fnτn) dτ1⋯dτn⎞⎠df1⋯dfn (5.9)


The right hand bracketed term in (5.9) is formally known in the literature as the

n-dimensional Fourier Transform of the nth order Volterra kernel and expressed as:

Hn(f1, . . . , fn) =∞

∫0

⋯∞

∫0

hn(τ1, . . . , τn) e−j2π(f1τ1+⋯+fnτn) dτ1⋯dτn (5.10)

Substituting (5.10) into (5.9) gives the response in terms of input signal components:

Hn

⎡⎢⎢⎢⎢⎣

∞

∫−∞

X(f) ej2πft df⎤⎥⎥⎥⎥⎦=

∞

∫−∞

⋯∞

∫−∞

X(f1)⋯X(fn) Hn(f1, . . . , fn) ej2π(f1+⋯+fn)t df1⋯dfn (5.11)

In terms of spectral regrowth, (5.11) tells us that input spectral components existing

at f1, . . . , fn will generate an output spectral component at f1+ . . .+fn with complex

amplitude:

Yn(f1, . . . , fn) = X(f1)⋯X(fn) Hn(f1, . . . , fn) (5.12)

It is important to highlight that Yn(f1, . . . , fn) in (5.12) represents the output con-

tribution from only one set of n inter-modulating spectral components. Since many

different sets [f1, . . . , fn] exist for which f1 + . . . + fn = f , the resultant output

spectral component at frequency f is obtained by accumulating all such individual

contributions:

Yn(f) =∞

∫−∞

∞

∫−∞

⋯∞

∫−∞

Yn(f −ϕ1, ϕ1 − ϕ2, . . . , ϕn−2 −ϕn−1, ϕn−1) dϕ1 dϕ2⋯dϕn−1

(5.13)

It follows that (5.13) is formally known in the literature as the Fourier Transform

of the output of the nth order Volterra operator.

The above discussion has introduced the Volterra Series and its inter-modulating

properties in the most general sense. In the next section, we apply this Volterra

model to the power amplifier at RF and subsequently derive an equivalent baseband

transmitter model.

5.2 RF and Baseband Transmitter Models

Figure 5.2 presents a block diagram of the transmitter as it would exist in physi-

cal implementation. We refer to this as an RF transmitter model since the power

amplifier processes an RF signal. In our predistortion work however, we desire an

equivalent transmitter model in which the power amplifier processes a baseband


signalmodulator

data in IQ frequencyupconverter

amplifier mask filters(t) s(t)A[ ⋅ ] M[ ⋅ ]

y(t)

Figure 5.2: RF transmitter model

signal. The reason being, the mathematical architecture of the power amplifier in

this equivalent baseband transmitter model dictates the mathematical architecture

of the corresponding baseband predistortion filter. In the following, we derive this

equivalent baseband transmitter model and hence present the required predistortion

filter architecture.2

Referring to Figure 5.2, the output of the signal modulator is represented by the

complex baseband signal s(t). Following IQ frequency upconversion3, the real RF

excitation signal s(t) can be represented as:

s(t) = s(t)ej2πft + s∗(t)e−j2πft (5.14)

where f represents the transmission carrier frequency and ∗ denotes complex con-

jugation. In this form, s(t) is commonly referred to as the complex baseband signal

envelope. In order to simplify notation, let:

s+1(t) = s(t)ej2πft and s−1(t) = s∗(t)e−j2πft (5.15)

Substituting back into (5.14), s(t) can now be represented as:

s(t) = s+1(t) + s−1(t) (5.16)

Here, the +1 and −1 subscripts are intended to signify the positive and negative

bandpass frequency characteristics of the respective signal components.

Representing the amplifier and mask filter in terms of the operators A[ ⋅ ] and

M[ ⋅ ] respectively, the transmitter output y(t) can be written as:

y(t) = M[ A[ s(t) ] ] (5.17)

Substituting (5.16) into (5.17) then gives:

y(t) = M[ A[ s+1(t) + s−1(t) ] ] (5.18)

2The power amplifier modeling community also shares this desire for baseband modeling but fordifferent reasons. In their case, a baseband transmitter model avoids the problems associated withhigh carrier frequency sampling rates [39].

3Ideal frequency upconversion is assumed here since mixer nonlinearities are considered negligiblecompared to those nonlinearities of the power amplifier.


Letting the amplifier A[ ⋅ ] be represented as a Volterra system, (5.18) can be ex-

pressed in terms of individual nth order Volterra operators:

y(t) = M[∞∑n=1

An[ s+1(t) + s−1(t) ] ] (5.19)

Expressing further in terms of n-linear Volterra operators:

y(t) = M

⎡⎢⎢⎢⎢⎣

∞∑n=1

( ∑a1=±1

⋯ ∑an=±1

An{sa1(t), . . . , san(t)} )⎤⎥⎥⎥⎥⎦

(5.20)

Since the bandpass mask filterM[ ⋅ ] limits spectral regrowth to the first-zone carrier

region, only those An{sa1(t), . . . , san(t)} terms in (5.20) for which

(a1 +⋯+ an = ±1) will actually be transmitted. If we let a+1 and a−1 represent the

vector subspaces of { [a1,⋯, an] } for which (a1 +⋯+ an = +1) and (a1 +⋯+ an = −1)respectively, then (5.20) can be rewritten as:

y(t) = M

⎡⎢⎢⎢⎢⎣

∞∑

odd n=1( ∑

a+1An{sa1(t), . . . , san(t)} + ∑

a−1An{sa1(t), . . . , san(t)} )

⎤⎥⎥⎥⎥⎦(5.21)

It is noted that even order n terms no longer exist in the outer summation and

hence the bandwidth of spectral regrowth components within the first-zone carrier

region must be odd multiples of the original modulation bandwidth. Assuming now

symmetric Volterra kernels, the inner summations of (5.21) can be simplified in

terms of binomial coefficients:

y(t) = M

⎡⎢⎢⎢⎢⎣

∞∑

odd n=1(

n

⌈n2 ⌉) An{s+1(t)1, . . . , s+1(t)⌈n

2⌉, s−1(t)1, . . . , s−1(t)⌊n

2⌋}

+ (n

⌈n2 ⌉) An{s−1(t)1, . . . , s−1(t)⌈n

2⌉, s+1(t)1, . . . , s+1(t)⌊n

2⌋}

⎤⎥⎥⎥⎥⎦(5.22)

Expressing the amplifier n-linear Volterra operators in functional form and expand-

ing the bandpass signals in terms of their complex baseband envelope according


to (5.15), then gives:

y(t) = M

⎡⎢⎢⎢⎢⎣

∞∑

odd n=1

ej2πft∞

∫0

⋯∞

∫0

an(τ1, . . . , τn) s(t−τ1)⋯s(t−τ ⌈ n2⌉) s∗(t−τ ⌈n

2⌉+1)⋯s∗(t−τn) dτ1⋯dτn

+ e−j2πft∞

∫0

⋯∞

∫0

a∗n(τ1, . . . , τn) s∗(t−τ1)⋯s∗(t−τ ⌈n2⌉) s(t−τ ⌈n

2⌉+1)⋯s(t−τn) dτ1⋯dτn

⎤⎥⎥⎥⎥⎦(5.23)

where:

an(τ1, . . . , τn) = (n

⌈n2 ⌉) an(τ1, . . . , τn) e

j2πf(−τ1−⋯−τ ⌈n2 ⌉+τ ⌈n2 ⌉+1

+⋯+τn)(5.24)

and an(τ1, . . . , τn) is the amplifier nth order Volterra kernel. To simplify (5.23), we

define the modified amplifier system operator:

A[s(t)] =∞∑

odd n=1An[s(t)] (5.25)

where the associated individual nth order nonlinear operators are defined as:

An[s(t)] =∞

∫0

⋯∞

∫0

an(τ1, . . . , τn) s(t−τ1)⋯s(t−τ ⌈ n2⌉) s∗(t−τ ⌈n

2⌉+1)⋯s∗(t−τn) dτ1⋯dτn

(5.26)

Substituting (5.26) back into (5.23) we obtain:

y(t) = M

⎡⎢⎢⎢⎢⎣

∞∑

odd n=1ej2πft An[s(t)] + e−j2πft A∗n[s(t)]

⎤⎥⎥⎥⎥⎦(5.27)

Regrouping the summation terms according to (5.25) then gives:

y(t) = M

⎡⎢⎢⎢⎢⎣A[s(t)] ej2πft +A∗[s(t)] e−j2πft

⎤⎥⎥⎥⎥⎦(5.28)

We immediately recognize (5.28) as representing the modified amplifier system op-

erator A[ ⋅ ] acting on the pure baseband signal modulation s(t) followed by an

ideal IQ modulator and mask filter as depicted in Figure 5.3. Since the modified

amplifier system operator is processing at baseband, this represents the baseband

transmitter model we so desire. For now obvious reasons, the functional form of this

modified amplifier system operator, (5.25) and (5.26), is known in the literature as


signalmodulator


amplifier mask filters(t)M[ ⋅ ]

y(t)A[ ⋅ ]

Figure 5.3: Baseband transmitter model

the Baseband Volterra Series. For the sake of consistency, we too will use this name

from herein.

Comparing Figures 5.2 and 5.3, we see that two differences exist between the RF

and baseband transmitter models. Firstly the frequency conversion and amplifica-

tion stages are swapped, which is to be expected since we have gone from an RF to

a baseband model. Secondly and most importantly, the amplifier system operator

changes from the pure Volterra Series to the Baseband Volterra Series. For com-

parison purposes, we repeat both of these operators below in terms of the general

signal x(t):

RF Amplifier Operator:

A[x(t)] =∞∑n=1

An[x(t)] (5.29)

where

An[x(t)] =∞

∫0

⋯∞

∫0

an(τ1, . . . , τn) x(t − τ1)⋯x(t − τn) dτ1⋯dτn (5.30)

Baseband Amplifier Operator:

A[x(t)] =∞∑

odd n=1An[x(t)] (5.31)

where

An[x(t)] =∞

∫0

⋯∞

∫0

an(τ1, . . . , τn) x(t−τ1)⋯x(t−τ ⌈ n2⌉) x∗(t−τ ⌈n

2⌉+1)⋯x∗(t−τn) dτ1⋯dτn

(5.32)

Since we are not intending to use a Model Based Derivation strategy for predistor-

tion filter parameter estimation, the RF to baseband kernel conversion an(τ1, . . . , τn)to an(τ1, . . . , τn) given by (5.24) is irrelevant in our predistortion work, though it

must be pointed out that this conversion leaves the baseband kernel being complex

in general. What is relevant in our work however is the removal of even order op-


signalmodulator


amplifier mask filterpredistorters(t)M[ ⋅ ]

y(t)A[ ⋅ ]P [ ⋅ ]

Figure 5.4: Baseband transmitter model with predistortion filter inserted

erators, compare (5.29) and (5.31), and also the introduction of conjugated product

terms, compare (5.30) and (5.32), because both have a direct bearing on the final

architecture of the predistortion filter.

5.3 Predistortion Filter Architecture

As discussed in Chapter 1, the digital predistortion process involves inserting a

nonlinear digital filter directly at the output of the signal modulator. This filter’s

transfer characteristic is designed to be the inverse of the power amplifier’s, thereby

creating an overall linear transmission path. Insertion of the digital predistortion

filter into the baseband transmitter model is shown in Figure 5.4.

It would now be logical to assume that the predistortion filter architecture with

the greatest potential for realizing this inverted transfer characteristic would be one

that shares the same general architecture as the cascaded amplifier operator A[ ⋅ ],thus capturing similar dynamic memory effects, while exhibiting a somewhat in-

verted set of kernel coefficients, thus performing the complementary dynamic com-

pensation. Based on this general thinking, we let the predistortion filter in our work

take on the same architecture as (5.31) and (5.32) but with a temporal discretiza-

tion since filtering is to be implemented digitally. An operator name change to P [ ⋅ ]along with a nonlinear order re-indexing from n to m is also required as presented

below:

Predistortion Filter Operator:

P [ s[k] ] =∞∑

odd m=1Pm[ s[k] ] (5.33)

where

Pm[ s[k] ] =∞∑i1=0

⋯∞∑im=0

pm[i1, . . . , im] s[k−i1]⋯s[k−i ⌈m2⌉] s∗[k−i ⌈m

2⌉+1]⋯s∗[k−im]

(5.34)

Here, k and i represent the discrete-time and delay variables respectively and pm[i1, . . . , im]represents the predistortion filter’smth order Baseband Volterra kernel. In Chapter 12,


we address the need to prune this predistortion filter kernel. Prior to this however,

we work in terms of the unpruned (5.33) and (5.34) in order to maximize insight

into algorithm development.

The important point to take away from this chapter is that whilst we judiciously

choose to model the RF power amplifier by the pure Volterra Series, the actual pre-

distortion filter ends up being modeled by the Baseband Volterra Series variant. This

result is attributed to the changing form of the power amplifier operator during the

RF to baseband transmitter model conversion.

Chapter 6

Digital Predistortion In The

Time-Domain

In this chapter we investigate the time-domain nonlinear interactions between pre-

distorter and amplifier with the goal of understanding how the predistorter is able

to cancel the nonlinear effects of the power amplifier and hence achieve linearization.

6.1 Intuitive Graphical Analysis

We begin with an intuitive graphical analysis of how the predistortion concept works

in the time-domain. Consider Figure 6.1 which presents the AM-AM characteristic

of a quasi-memoryless power amplifier1. Here it can be seen that as input signal

amplitude increases, output signal amplitude starts to compress before eventually

saturating, leaving the amplifier with a non-constant gain and a precise linear work-

ing region.

Without predistortion, the amplifier’s input signal is scaled such that its ampli-

tude generally remains within the linear region as depicted by the left most Proba-

bility Density Function (PDF) at the bottom of the figure. In this case, distortion

is avoided but the amplifier is running inefficiently with severe output back off2.

Consider now the scenario of predistortion. The predistorter’s input signal is

scaled such that its peak amplitude equals CI as depicted by the middle PDF at

the bottom of the figure. For those upper amplitudes now within the amplifier’s

nonlinear region (AI to CI), the predistorter performs expansion in order to com-

pensate for the amplifier’s impending compression. For example, the predistorter

1The same general concepts about to be covered extend to the dynamic case but are easier tocomprehend without AM-AM hysteresis.

2 [252] states that Crest Factor is commonly used to determine the amplifier’s output back offbut believes this is not always the correct method since it is the statistical envelope distributionthat will determine average distortion. We agree with this line of thinking.

54

Input Amplitude

Out

put A

mpl

itude

Desired Linear ResponseSaturation

MaximumCorrectable

Input

Input ExpansionDue To Predistortion

Output ExpansionDue To Predistortion

Amplifier’s AM-AMCharacteristic

PDF of inputwithout predistortion

PDF ofpredistorter

input

Linear Region Nonlinear Region

PDF ofpredistorter

output

AI BIBExpanded

I

CI CExpandedI

ALinearO

BCompressedO

BLinearO

CCompressedO

CLinearO

X

Y Z

Figure 6.1: Graphical analysis of digital predistortion in the time-domain

will expand input BI to BExpandedI so that the amplifier output also expands from

BCompressedO to the desired BLinear

O . In this way, the amplifier is able to operate effi-

ciently within its nonlinear region yet still appear linear. Predistortion does have its

practical limits however with CI representing the maximum correctable input and

hence upper limit to linearization. Beyond this point, signal expansion by the predis-

torter is capped by full output saturation. Overall, this gives the predistorted power

amplifier a maximally hard saturation characteristic represented by line segments

X-Y-Z of Figure 6.1 [131,153].

It should not be forgotten that AM-PM related distortion is also corrected for

during the predistortion process, however it is this signal expansion concept which

most intuitively represents the workings of the digital predistortion filter.

CHAPTER 6. DIGITAL PREDISTORTION IN THE TIME-DOMAIN 56

6.2 Mathematical Operator Analysis

We now turn attention to a mathematical operator analysis of the interactions be-

tween predistorter and power amplifier. In the previous chapter, we derived the

baseband equivalent transmitter model and inserted the predistortion filter at the

output of the signal modulator according to Figure 5.4, page 52. The cascaded pre-

distorter P [ ⋅ ] and power amplifier A[ ⋅ ] were both represented by the Baseband

Volterra Series, now assumed to be discretized. Figure 6.2 presents this cascade

along with a unique representation of its output which we have pioneered and refer

to as the Distortion Array.

Effectively, the Distortion Array is a graphical organizing tool for keeping track

of those nonlinear distortion components generated by the cascade. Its elements are

either blank or ticked. A blank element signifies nil distortion whilst a ticked element

signifies one or more distortion components. Ticks are also color coded to classify

the origin of components. Each row of the array is associated with components

generated by the corresponding in-feeding amplifier operator whilst each column of

the array is associated with components of equivalent order as labeled. The output

of the nth order amplifier operator (n odd) is given by:

An[P [ s[k] ] ] = An

⎡⎢⎢⎢⎢⎣

∞∑

odd m=1Pm[ s[k] ]

⎤⎥⎥⎥⎥⎦(6.1)

Expanding the right hand side of (6.1) in terms of n-linear operators then gives:

An[P [ s[k] ] ] =∞∑

odd m1 =1⋯

∞∑

odd mn =1An{ Pm1[ s[k] ], . . . ,Pmn[ s[k] ] } (6.2)

Each n-linear operator An{Pm1[ s[k] ], . . . ,Pmn[ s[k] ] } in (6.2) represents a single

distortion component, which by definition (5.6), is of nonlinear order (m1+⋯+mn).Each of these components make up the Distortion Array row being fed by amplifier

operator An[ ⋅ ]. Since both n and m are odd, each component will be of odd

nonlinear order. From (6.2), we can make some very important observations:

1. The minimum order of distortion generated by An[ ⋅ ] is n, corresponding to

component An{ P 1[ s[k] ], . . . ,P 1[ s[k] ] }. In terms of the Distortion Ar-

ray, this means that the row being fed by An[ ⋅ ] commences at the nth order

column, and a leading diagonal forms across the array. Furthermore, since

digital predistortion does not attempt to compensate for an amplifier’s lin-

ear distortion, P 1[ ⋅ ] can be assumed transparent, that is its kernel is the

unit impulse, and P 1[ s[k] ] = s[k]. In effect, these minimum order lead-

ing diagonal components of the array then represent pure amplifier distortion


s[k]

P 1[ ⋅ ]

P 3[ ⋅ ]

P 5[ ⋅ ]

P 7[ ⋅ ]

P 9[ ⋅ ]

⋮

A1[ ⋅ ]

A3[ ⋅ ]

A5[ ⋅ ]

A7[ ⋅ ]

A9[ ⋅ ]

⋮

1st 3rd 5th 7th 9th 11th 13th

q[k]

✓ ✓ ✓ ✓ ✓ ✓ ✓

✓ ✓ ✓ ✓ ✓

✓ ✓ ✓ ✓

✓ ✓ ✓

✓ ✓

✓

✓

✓

✓

Order of Nonlinear Components

Figure 6.2: Predistorter-Amplifier cascade (left) and Distortion Array (right)

An{ s[k], . . . , s[k] } = An[ s[k] ]. To reflect this, the leading diagonal ticks are

colored green, the same color as the amplifier operator blocks. These compo-

nents need to be canceled as per the original linearization problem.

2. Since A1[ ⋅ ] is a linear operator, its output components are merely linearly

filtered predistorter components. For the purposes of this analysis, we can

assume negligible linear amplifier distortion and as such approximate the out-

put of A1[ ⋅ ] to be the predistorter components scaled by the gain G of the

amplifier. Because of this relationship with the predistorter, ticks along the

first row of the Distortion Array are colored red, the same color as the predis-

torter operator blocks. As will become evident shortly, since these components

avoid nonlinear amplifier inter-modulation, they will be used as the tools for

canceling other Distortion Array components of equivalent order.

3. We refer to the elements of the Distortion Array that aren’t on the leading

diagonal (green) or in the first row (red) as parasitic elements since they rep-

resent unwanted by-products of inserting the predistortion filter, specifically

inter-modulation components An{Pm1[ s[k] ], . . . ,Pmn[ s[k] ] } where n ≠ 1

and (m1,⋯,mn) don’t all equal 1. These components are neither pure ampli-

fier nor predistorter. Parasitic element ticks are colored black in the Distortion

Array to signify their unwanted nature.


s[k]

P 1[ ⋅ ]

P 3[ ⋅ ]

P 5[ ⋅ ]

P 7[ ⋅ ]

P 9[ ⋅ ]

⋮

A1[ ⋅ ]

A3[ ⋅ ]

A5[ ⋅ ]

A7[ ⋅ ]

A9[ ⋅ ]

⋮


q[k]

✓ ✓ ✓ ✓ ✓ ✓ ✓

✓ ✓ ✓ ✓ ✓

✓ ✓ ✓ ✓

✓ ✓ ✓

✓ ✓

✓

✓

✓

✓


Figure 6.3: Parasitic components generated by Pm[ ⋅ ]

Parasitic components generated by Pm[ ⋅ ] (m ≥ 3) will be of nonlinear order

greater than m. In the Distortion Array, these parasitic components will

be bounded to the left by a diagonal line 1) running parallel to the green

leading diagonal and 2) passing through the mth order element of the first

row. Figure 6.3 demonstrates this by associating each parasitic element of the

Distortion Array with its generating predistorter operator block by way of a

small colored box. Those parasitic elements which possess more than one small

colored box subsequently represent multiple parasitic components of different

origin. For example the two components:

A3{ P 1[ s[k] ], P 3[ s[k] ], P 3[ s[k] ] } (6.3)

A3{ P 1[ s[k] ], P 1[ s[k] ], P 5[ s[k] ] } (6.4)

are both 7th order parasitic components represented by the single element in

the 7th order column of the row being fed by A3[ ⋅ ]. (6.3) is generated by

P 3[ ⋅ ] (small blue box) while (6.4) is generated by P 5[ ⋅ ] (small yellow box).


6.3 Ideal Predistorter Operator

To take the next step in understanding how each predistorter operator is configured

to linearize the power amplifier, we must view the predistorter-amplifier cascade in

terms of its equivalent cascade nonlinearity Q[ ⋅ ]. Figure 6.4 presents this equiva-

lent cascade nonlinearity in terms of its individual operators Ql[ ⋅ ] along with the

original representation of the predistorter-amplifier cascade and Distortion Array.

We immediately see that Ql[ s[k] ] is the sum of components represented by the lth

order column of the Distortion Array :

Q1[ s[k] ] = Gs[k] (6.5)

Q3[ s[k] ] = A3[ s[k] ] +GP 3[ s[k] ] (6.6)

For odd l ≥ 5 Ql[ s[k] ] = Al[ s[k] ] +GP l[ s[k] ] +U l[ s[k] ] (6.7)

In each of (6.5), (6.6) and (6.7), terms are color coded in accordance with the

elements of the Distortion Array to which they correspond; namely amplifier (green),

predistorter (red) and parasitic (black). In (6.7), U l[ s[k] ] represents the set of

Unwanted lth order parasitic components:

U l[ s[k] ] =l−2∑

odd n=3

⎡⎢⎢⎢⎢⎣∑mnl

An{Pm1[ s[k] ],⋯,Pmn[ s[k] ] }⎤⎥⎥⎥⎥⎦

(6.8)

Here, mnlis the vector subspace of { [m1,⋯,mn] } for which (m1 + ⋯ +mn = l).

Consistent with our earlier findings, we see from (6.8) that lth order parasitic com-

ponents are generated by lower order predistorter operators.3

It can now be seen from (6.6) that Q3[ s[k] ] can be eliminated and hence 3rd order

amplifier linearization achieved if:

P 3[ s[k] ] =−A3[ s[k] ]

G(6.9)

Similarly from (6.7), for odd l ≥ 5, Ql[ s[k] ] can be eliminated and hence lth order

amplifier linearization achieved if:

P l[ s[k] ] =−(Al[ s[k] ] +U l[ s[k] ] )

G(6.10)

3For clarification, since each n-linear operator An{⋯} of (6.8) can be represented as an accu-mulation of standard operators An[ ⋅ ] acting on partial input accumulations [240, 248], U l[ ⋅ ] canbe considered a Baseband Volterra operator in its most fundamental form.


s[k]

s[k]

P 1[ ⋅ ]

P 3[ ⋅ ]

P 5[ ⋅ ]

P 7[ ⋅ ]

P 9[ ⋅ ]

⋮⋮

⋮

⋮

A1[ ⋅ ]

A3[ ⋅ ]

A5[ ⋅ ]

A7[ ⋅ ]

A9[ ⋅ ]

⋮


q[k]

q[k]

✓ ✓ ✓ ✓ ✓ ✓ ✓

✓ ✓ ✓ ✓ ✓

✓ ✓ ✓ ✓

✓ ✓ ✓

✓ ✓

✓

✓

✓

✓

Q1[ ⋅ ]

Q3[ ⋅ ]

Q5[ ⋅ ]

Q7[ ⋅ ]

Q9[ ⋅ ]

Figure 6.4: Equivalent cascade nonlinearity Q[ ⋅ ]


So for odd l ≥ 5, not only is P l[ ⋅ ] required to cancel Al[ ⋅ ] (the original lth or-

der amplifier nonlinearity), but it’s also required to cancel the lth order parasitic

components generated by lower order predistorter operators.

The implication of this last fact is that whilst any single order of amplifier lin-

earization can be achieved according to (6.9) and (6.10), such a linearization will

become obsolete if a lower order predistorter operator changes for any reason. It

follows that if one plans to linearize the amplifier up to lth order, computation of

predistorter operators must occur in ascending order up to lth order.

6.4 Theoretical Limits of Digital Predistortion

Despite the fact that any single order of amplifier linearization is achievable accord-

ing to (6.9) and (6.10), entire linearization is not theoretically possible. This is quite

simply because every time a predistorter operator is computed to eliminate amplifier

and / or parasitic distortion of equivalent order, it produces higher order parasitic

components in the process. So no matter how high an order one wishes to linearize

up to, at least parasitic distortion will remain.

Thankfully, we can assume higher order parasitic components are 1) generally

uncorrelated and therefore don’t add constructively and 2) are of progressively lower

power. This means that despite not ever being able to theoretically achieve entire

linearization, parasitic distortion can be reduced with each higher order of lineariza-

tion. In practice, predistortion is performed for 3rd order up to some finite maximum

order. Choosing this maximum order is a trade off between amplifier / parasitic dis-

tortion levels (keeping in mind regulatory spectral mask and terminal sensitivity

requirements) and predistorter computational complexity. This trade off will be

discussed in more detail in Chapter 8.

Before leaving this chapter, it is worth noting that this operator analysis of digital

predistortion has strong links with the P th Order Inverses technique, a Model Based

Derivation strategy first introduced within the literature review. The Distortion Ar-

ray was in fact developed here as a means of more intuitively portraying the central

concepts of this technique which are often overshadowed by mathematical rigor. This

mathematical rigor is a result of expanding the U l[ ⋅ ] operator of (6.8) in terms of

its standard Baseband Volterra operators. The aim of this expansion is to derive

the exact form of the linearizing predistorter operators (6.10). Such an exact form

is unnecessary in our work however since we intend using a Self-Learning strategy

rather than Model Based Derivation to compute predistorter kernels. In other words,

in our case, concept is more important than precise mathematical form. Discussion

of the intended Self-Learning strategy begins in Chapter 9.

Chapter 7

Digital Predistortion In The

Frequency-Domain

In the previous chapter, we analyzed the time-domain nonlinear interactions existing

between predistorter and amplifier and were able to gain an intuitive understanding

of how the predistorter achieves linearization. In this chapter we extend this insight

to the frequency-domain in order to show how the amplifier output spectrum behaves

during the linearization process.

Predistortion in the frequency-domain is best viewed in terms of the equiva-

lent cascade nonlinearity Q[ ⋅ ], introduced in the previous chapter and repeated in

Figure 7.1. Here, individual operators Ql[ ⋅ ] are represented in terms of their con-

stituent amplifier, predistorter and parasitic distortion operators according to (6.5),

(6.6) and (6.7). It is worth noting that since both predistorter and amplifier are

modeled as Baseband Volterra systems, the equivalent cascade nonlinearity Q[ ⋅ ]can also be modeled as such.

Consider now the general lth order cascade nonlinearity Ql[ ⋅ ] with input random

process s[k]. Let s[k] be representative of our OFDM/CDMA target signal mod-

ulations, that is, complex baseband with continuous-time bandwidth B centered at

0Hz. This is depicted by the power spectral density Sss(f) in the bottom left of

Figure 7.1. A defining characteristic of nonlinear systems is spectral regrowth. Since

l-fold time-domain multiplication within Ql[ ⋅ ]’s representative Baseband Volterra

Series is equivalent to l-fold frequency-domain convolution [307], Ql[ ⋅ ] will generatea complex baseband output process ql[k] with continuous-time bandwidth lB cen-

tered at 0Hz. This is depicted by the power spectral density Sqlql(f) in the bottom

right of Figure 7.1.

62

s[k]

Q1[ ⋅ ] = Gs[k]

Q3[ ⋅ ] = A3[ ⋅ ] +GP 3[ ⋅ ]

Q5[ ⋅ ] = A5[ ⋅ ] +GP 5[ ⋅ ] +U5[ ⋅ ]

Q7[ ⋅ ] = A7[ ⋅ ] +GP 7[ ⋅ ] +U7[ ⋅ ]

Q9[ ⋅ ] = A9[ ⋅ ] +GP 9[ ⋅ ] +U9[ ⋅ ]

⋮

q1[k]

q3[k]

q5[k]

q7[k]

q9[k]

q[k]

Sss(f) Sqlql(f)

ff

B lB

Figure 7.1: Equivalent cascade nonlinearity Q[ ⋅ ]

CHAPTER 7. DIGITAL PREDISTORTION IN THE FREQUENCY-DOMAIN 64

While l-fold spectral regrowth and a 0Hz center frequency is guaranteed, the

level of Sqlql(f) will ultimately depend on the makeup of the nonlinearity Ql[ ⋅ ]and therefore the current state of the predistorter. Since practical predistortion

is generally performed for 3rd order up to some finite maximum order (refer to

Section 6.4), we will examine the levels of Sqlql(f) at each progressively higher

predistortion state and draw conclusions.

First consider Sqlql(f) prior to predistortion. In this state, Ql[ ⋅ ] = Al[ ⋅ ] and

output distortion is pure amplifier distortion. This is shown in Figure 7.2 for

a real 25Watt Class-AB push-pull power amplifier with DAB signal modulation

(B = 1.537MHz). It is noted that higher orders of distortion are present within this

figure but their spectral envelopes are hidden beneath the noise floor.

Ref Lvl

11 dBm

Ref Lvl

11 dBm

RBW 30 kHz

VBW 300 kHz

SWT 2 s

RF Att 30 dB

1.2 MHz/Center 200 MHz Span 12 MHz

-4 dB Offset

Mixer -20 dBm

A

LN

Unit dBm

2RM2VIEW

-80

-70

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-30

-20

-10

0

-89

11

Sq1q1(f)

Sq3q3(f)

Sq5q5(f)

Sq7q7(f)

Figure 7.2: Power spectra Sqlql(f) prior to predistortion for a real 25Watt Class-ABpush-pull power amplifier with DAB signal modulation


Now assume 3rd order predistortion is performed and consider the level of Sqlql(f).Recall from Chapter 6 and the Distortion Array that 3rd order predistortion elimi-

nates Q3[ ⋅ ], and hence Sq3q3(f), but in the process generates higher order parasitic

distortion. It follows that Ql[ ⋅ ] for l ≥ 5 will change from pure amplifier distortion

to Ql[ ⋅ ] = Al[ ⋅ ] +U l[ ⋅ ] and Sqlql(f) will experience a slight growth. This spectral

behavior is illustrated in Figure 7.3.

Power

Noise Floor

Sq1q1(f)

Sq5q5(f)

Sq7q7(f)

f

B B B BBBB

Sq9q9(f)

Figure 7.3: Power spectra Sqlql(f) after 3rd order predistortion (solid traces).Dashed traces represent levels prior to predistortion and arrows highlight parasiticgrowth. 3rd order distortion is totally eliminated. Higher orders of distortion arepresent beneath the noise floor but aren’t shown to avoid clutter.


A real example of 3rd order predistortion is also presented in Figure 7.4 for a 25Watt

Class-AB push-pull power amplifier with DAB signal modulation. It can be seen

here that after 3rd order predistortion, the distortion characteristic is less rounded;

indicating that only higher order distortion remains. Also, this higher order dis-

tortion becomes visible above the noise floor where it wasn’t visible before; a clear

demonstration of parasitic growth. It must be noted that Figure 7.4 exhibits a 10 dB

higher noise floor compared to Figure 7.2, despite both being associated with the

same amplifier and signal modulation. This is due to the finite resolution of recon-

struction DACs and the heavier scaling of their input signals to accommodate the

impending Crest Factor growth of predistortion.

RBW 30 kHz

VBW 300 kHz

SWT 2 s

RF Att 30 dB

Mixer -20 dBm

A

LN

2RM

Unit dBm

1VIEW

2VIEW

1RM

Ref Lvl

11 dBm

Ref Lvl

11 dBm


1 dB Offset *

-80

-70

-60

-50

-40

-30

-20

-10

0

-89

11

Before Predistortion(only 3rd order distortion visible above noise floor)

After 3rd Order Predistortion(only higher order distortion remains)

Growth In Higher Order Distortion(now visible above noise floor)

Figure 7.4: Power spectra before and after 3rd order predistortion for a real 25WattClass-AB push-pull power amplifier with DAB signal modulation


Now assume 3rd order predistortion is immediately followed by 5th order predistor-

tion and consider the level of Sqlql(f). Recall from Chapter 6 and the Distortion

Array that 5th order predistortion:

1. eliminates Q5[ ⋅ ] and hence Sq5q5(f)

2. has no effect on Ql[ ⋅ ] for l < 5 and therefore Sq3q3(f) remains eliminated

3. generates higher order parasitic distortion which causes Sqlql(f) to grow for l > 5.

This spectral behavior is illustrated in Figure 7.5 in a similar manner to Figure 7.3.

Noise Floor

Power

Sq1q1(f)

Sq7q7(f)

f

B B B BBBB

Sq9q9(f)

Figure 7.5: Power spectra Sqlql(f) after 5th order predistortion (solid traces).Dashed traces represent levels prior to predistortion, dotted traces represent lev-els after 3rd order predistortion and arrows highlight parasitic growth. 3rd and 5th

order distortion are totally eliminated. Higher orders of distortion are present be-neath the noise floor but aren’t shown to avoid clutter.


This same analysis can be continued for even higher orders of predistortion with the

same conclusions drawn in each case. That is, lth order predistortion will:

1. eliminate Ql[ ⋅ ] and therefore Sqlql(f)

2. have no effect on Qj[ ⋅ ] and therefore Sqjqj(f) for j < l

3. generate higher order parasitic distortion and therefore increase Sqkqk(f) for k > l

It is important not to be misled by Dot-Point 3 above when considering the resultant

spectral distortion. While each progressive order of predistortion will generate higher

order parasitics, this growth in distortion is significantly less than the reduction in

distortion caused by the corresponding Dot-Point 1. Hence with each progressive

order of predistortion, the resultant spectral distortion does indeed reduce.1

Based on the preceding analysis, the three important points to take away from

this chapter are summarized below:

1. Equivalent cascade nonlinearity Ql[ ⋅ ] causes l-fold spectral regrowth.

2. Spectral regrowth from each equivalent cascade nonlinearity is collocated in

frequency, specifically centered at 0Hz.

3. With each additional order of predistortion, the resultant spectral distortion

is reduced and transmitter linearization is further enhanced.

In the preceding analysis, we have been focused on the output power spectrum

of the equivalent cascade nonlinearity since this is what is ultimately transmitted.

Before leaving this chapter however, it is worthwhile considering the output power

spectrum of the predistorter, if only to gain intuitive insight.

For comparison purposes, Figure 7.6a presents the output power spectrum of

the equivalent cascade nonlinearity before and after a full predistortion is performed.

For consistency, this is for the same 25Watt Class-AB push-pull power amplifier and

DAB signal modulation as used in earlier figures. Here we can see that predistortion

successfully reduces distortion to the noise floor limit.

Figure 7.6b then presents the output power spectrum of the predistorter before

and after this same full predistortion. We can see here the distortion that is created

by the predistorter in order to perform the linearization. Comparing Figures 7.6a

and 7.6b, we also observe that this linearizing distortion is larger than the original

amplifier distortion; a clear demonstration that additional distortion is required to

cancel the self-generated parasitic distortion.

1The level of parasitic growth illustrated in Figures 7.3 and 7.5 is indicative only.

LD

Ref Lvl

16 dBm

Ref Lvl

16 dBm

RBW 30 kHz

VBW 300 kHz

SWT 2 s

RF Att 50 dB

500 kHz/Center 200 MHz Span 5 MHz

1 dB Offset

Mixer -40 dBm

A

2RM

3RM

Unit dBm

2VIEW

3VIEW

-80

-70

-60

-50

-40

-30

-20

-10

0

10

-84

16

before

after

Equivalent Cascade Output

(a) Output power spectrum of equivalent cascade nonlinearity before predistortion (red)and after predistortion (green).

Ref Lvl

13.5 dBm

Ref Lvl

13.5 dBm

RBW 30 kHz

VBW 300 kHz

SWT 2 s

RF Att 0 dB


53.5 dB Offset

Mixer -40 dBm

A

4RM

Unit dBm

1VIEW

4VIEW

1RM

*

-80

-70

-60

-50

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-20

-10

0

10

-86.5

13.5

before after

Predistorter Output

(b) Output power spectrum of predistorter before predistortion (magenta) and after pre-distortion (blue).

Figure 7.6: Comparison of equivalent cascade and predistorter output power spectra

Chapter 8

Maximum Order Of

Predistorter Nonlinearity

Now that we’ve covered predistortion in both the time and frequency domain, and

understand the interaction between predistorter and amplifier, we are in a position

to analytically choose the maximum order of predistorter nonlinearity. As already

touched on in Section 6.4, this choice is a trade off between linearization performance

and predistorter computational complexity. That is, the greater the maximum non-

linearity, the better the linearization performance but the higher the predistorter

computational complexity. The aim is thus to find a suitable balance.

In practice, an amplifier’s 3rd order nonlinearity is significantly larger than its

higher order nonlinearities. This was demonstrated in Figure 7.2 (page 64) for a real

25Watt Class-AB power amplifier with DAB signal modulation. It logically follows

from Chapter 6 and the Distortion Array that the predistorter’s largest nonlinearity

must also be 3rd order if linearization is to be effective. This ultimately means that

the parasitic distortion components generated by the large 3rd order predistorter

nonlinearity P 3[ ⋅ ] inter-modulating with the large 3rd order amplifier nonlinearity

A3[ ⋅ ] will dominate all other parasitic and amplifier nonlinearities above 3rd order.

To be precise, these parasitic distortion components are:

A3{ P 3[ s[k] ], s[k], s[k] } 5th order (8.1)

A3{ P 3[ s[k] ], P 3[ s[k] ], s[k] } 7th order (8.2)

A3{ P 3[ s[k] ], P 3[ s[k] ], P 3[ s[k] ] } 9th order (8.3)

For graphical reference, Figure 8.1 presents the Distortion Array with these domi-

nant amplifier, predistorter and parasitic elements marked.

70

CHAPTER 8. MAXIMUM ORDER OF PREDISTORTER NONLINEARITY 71

dominantpredistorter

element dominantparasiticelements

dominantamplifierelement

s[k]

P 1[ ⋅ ]

P 3[ ⋅ ]

P 5[ ⋅ ]

P 7[ ⋅ ]

P 9[ ⋅ ]

⋮

A1[ ⋅ ]

A3[ ⋅ ]

A5[ ⋅ ]

A7[ ⋅ ]

A9[ ⋅ ]

⋮


q[k]

✓ ✓ ✓ ✓ ✓ ✓ ✓

✓

✓

✓

✓

✓

✓

✓

✓

✓

✓✓

✓✓✓

✓✓✓✓


Figure 8.1: Dominant elements of the Distortion Array

It can be seen that if the maximum predistorter nonlinearity is chosen to be

less than 9th order, some of the dominant parasitic components will remain and

linearization performance will be jeopardized. To avoid this, maximum predistorter

order must be ≥ 9.

Considering the trade off between linearization performance and computational

complexity, we favor the lower end of this range and choose not to predistort higher

than 9th order as this will ensure computational processing does not become a lim-

iting factor in our work.

Chapter 9

SPFL Strategy With The New

WACP Optimization Objective

As stated in the Literature Review, the design and analysis of a digital predistortion

system is framed by two fundamental questions:

1. What architecture will the predistortion filter take?

2. How will the characterizing parameters of this filter be estimated?

In terms of Question 1, we know from Chapters 5 – 8 that the predistortion filter

will be modeled as a (yet to be pruned) Baseband Volterra Series consisting of

odd order nonlinearities up to 9th order. In terms of Question 2, we know from our

Statement of Research that predistortion filter parameter estimation will be based on

the Spectral Power Feedback Learning (SPFL) strategy. In this chapter, we present

a formal mathematical framework for this learning strategy and go on to propose a

new spectral power feedback objective more suited to current and future wideband

applications.

9.1 Mathematical Framework of SPFL Strategy

The Spectral Power Feedback Learning (SPFL) strategy, as presented in Figure 9.1,

models predistortion filter parameter estimation as a generic single objective math-

ematical optimization problem. In the applied mathematics community, such a

problem is formally defined as follows [200]:

Given a set of variables h plus a dependent function of those variables

B(h) = b, the goal of single objective mathematical optimization is to find

the optimal variable values ho which minimize the dependent function;

B(ho) = bmin. The dependent function to be minimized is referred to as

the objective function.

72

CHAPTER 9. THE NEW WACP OPTIMIZATION OBJECTIVE 73

signalmodulator


filter


filters


upconverter

poweramplifier

spectralpower

measurement

objectivefunction

formulation

tx out


Figure 9.1: Predistortion filter parameter estimation based on the SPFL strategy

In accordance with this definition and in the context of the SPFL strategy:

• The predistortion filter’s characterizing parameters are interpreted as the set

of variables h to be optimized. Since the predistortion filter is modeled as the

Baseband Volterra Series, these characterizing parameters are in fact a set of

Volterra kernels. Specifically, with the predistortion filter containing only odd

order nonlinearities up to 9th order, the optimization vector space h can be

represented as:

h = [p3 ∣ p5 ∣ p7 ∣ p9 ] (9.1)

where for m = 3, 5, 7, 9 :

pm = { pm[i1, . . . , im] ∀ causal [i1, . . . , im] } (9.2)

Here, pm[i1, . . . , im] represents the mth order predistortion filter kernel and

causality implies nonnegative delay variables.1

• Since transmitter linearization is ultimately dependent on these predistortion

filter kernels and hence h, any single measure of output spectral distortion can

be interpreted as the objective function B(h) = b to be minimized.

In summary, the optimal predistortion filter parameters ho must be found which

minimize the measure of spectral distortion, B(ho) = bmin, and ultimately linearize

the transmitter. Here, the objective is assumed nonconvex with multiple local min-

ima [78,265]. Also, optimizations are performed numerically since the objective has

no closed form; needing to be physically measured rather than formulated.

1(9.2) will be refined in Chapter 12 after the predistorter Baseband Volterra Series is pruned.


9.2 The New WACP Optimization Objective

In the literature to date, the SPFL strategy has only been reported with:

• linear signal modulation (QAM/QPSK)

• a very basic predistortion filter (up to 3 tunable parameters)

• the measure of spectral distortion being the power detected within a small

bandpass region of the transmitter’s adjacent channel [107,262–265]

Since our application of the SPFL strategy is to wider band OFDM/CDMA systems

exhibiting memory, our predistortion filter (specifically a pruned Baseband Volterra

Series) will contain a greater number of tunable parameters than that reported

above. With this increase in both modulation bandwidth and optimizer degrees of

freedom comes the regionalized behavior of the adjacent channel distortion spectrum.

By this it is meant that different regions of the adjacent channel distortion spectrum

will generally behave differently during the predistortion process. For example,

distortion reduction in one spectral region may be accompanied by no change, or

even worse, distortion growth in another spectral region. This ultimately means

that the regionalized measure of spectral distortion reported above is incapable of

conveying complete adjacent channel behavior and in fact encourages the optimizer

to tune the predistorter to solely reduce power in that region only; an unsatisfactory

result. It follows that with this push to wider band applications comes the need for

a more sophisticated multi-region distortion measure.

The multi-region distortion measure that we develop and propose in this research

is called the Weighted Adjacent Channel Power (WACP) and is presented below:

WACP = ∫LAC

W (f)P (f)df + ∫UAC

W (f)P (f)df (9.3)

Here, W (f) represents a nonnegative, frequency dependent weighting function,

specifically set to be a nonincreasing function of ∣f − fE ∣ where fE is the closest

(upper or lower) transmission band Edge frequency. P (f) represents the trans-

mitter output Power Spectral Density as a function of frequency and LAC /UAC

represent the integration domains of the Lower /Upper Adjacent Channel frequen-

cies respectively. It can be seen that standard Adjacent Channel Power (ACP) is

just the specific case of WACP for the constant weighting function W (f) = 1.

We propose the WACP objective over the more obvious standard ACP due to the

inaccuracies of the predistortion filter model. That is, if one uses a primitive Volterra

Series pruning strategy2 or incorrectly estimates predistortion filter memory, then

2Pruning strategies will be discussed in Chapter 12


BeforePredistortion

After IdealPredistortion

After Predistortion UsingStandard ACP Objective

Lower AdjacentChannels

TransmissionBand

Upper AdjacentChannels

ffCarrier

Figure 9.2: Power amplifier output spectra, before predistortion (red), after predis-tortion using a standard ACP objective (blue) and after ideal predistortion (green)

an optimizer using the standard ACP objective will tend to reduce outer adjacent

channel distortion in favor of co-channel and inner adjacent channel distortion. As

depicted in Figure 9.2, this is an unfavorable outcome since the final output mask

filter cannot be relied upon to remove distortion close to band edges given its finite

rolloff, plus co-channel distortion degrades BER performance. In either case, the

optimizer simply cannot cope with the inaccuracies of the predistortion filter model.

This behavior was experienced first hand in our very early experiments carried out on

the laboratory transmitter testbed. Although predistortion filter model inaccuracies

can be mitigated with appropriate design, they can never be totally eliminated and

hence an ACP objective function is considered too unreliable.

In theory, adding the frequency dependent weighting function W (f) to the stan-

dard ACP accumulation gives the objective function the added ability to discrimi-

nate between spectral distortion components, rather than treating them all equally

in the accumulation. Specifically setting W (f) to be a nonincreasing function of

∣f − fE ∣ (spectral distance from the closest transmission band edge) then forces the

optimizer to place greater emphasis on those previously neglected inner frequency

components. In essence, the optimizer becomes more robust in the presence of

predistortion filter model inaccuracies.

Many frequency dependent weighting functions W (f) that fit the requirement of

being nonincreasing functions of ∣f − fE ∣ can be derived, the two simplest being the


Linear

Quadratic Higher Order


TransmissionBand


f0f0 fEfE

WEWE

ffCarrier

Figure 9.3: Linear, quadratic and higher order weighting functions plotted withrespect to the allocated transmission band.

linear and quadratic functions which are formulated in (9.4) and (9.5) respectively:

W (f) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

0 for ∣f − fE ∣ > ∣f0 − fE ∣

⎛⎝−WE ∣f − fE ∣

∣f0 − fE ∣⎞⎠+WE for ∣f − fE ∣ ≤ ∣f0 − fE ∣

(9.4)

W (f) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

0 for ∣f − fE ∣ > ∣f0 − fE ∣

⎛⎝WE ∣f − fE ∣2

∣f0 − fE ∣2⎞⎠−⎛⎝2WE ∣f − fE ∣

∣f0 − fE ∣⎞⎠+WE for ∣f − fE ∣ ≤ ∣f0 − fE ∣

(9.5)

For graphical reference, these functions are presented in Figure 9.3, along with an

indicative higher order weighting function, demonstrating their relationship with

respect to the allocated transmission band. As can be seen, WE represents the

weighting function amplitude at the allocated transmission band edge frequency fE

whilst f0 represents the adjacent channel frequency at which the weighting falls to

zero. In accordance with the integration domains of (9.3), W (f) is not defined

within the allocated transmission band as spectral distortion power P (f) cannot be


measured here. In all situations, f0 must be chosen to span all distortion components

appearing above the indicated noise floor.

In practice, choosing a suitable weighting function amplitude WE is a balance

between the two extremes of under-weighting and over-weighting :

• Under-weighting is the scenario that arises when the weighting function ampli-

tude WE is insufficient and the optimizer behaves as if no weighting function

exists at all. That is, identical to using a standard ACP objective, the opti-

mizer tends to reduce outer adjacent channel distortion in favor of co-channel

and inner adjacent channel distortion.

• Over-weighting, in direct contrast, is the scenario that arises when the weight-

ing function amplitude WE is excessive to the point where the optimizer tends

to reduce inner adjacent channel distortion in favor of co-channel and outer

adjacent channel distortion. This is undesirable with co-channel distortion

degrading BER performance.

Both scenarios are presented in Figure 9.4. As one would expect, the initial stages

of predistortion are particularly sensitive to under-weighting since inner adjacent

channel distortion already dominates outer adjacent channel distortion. In direct

contrast, the latter stages of optimization are particularly sensitive to over-weighting

since inner and outer adjacent channel distortion is generally comparable at this

BeforePredistortion

After IdealPredistortion

After Predistortion WithUnder-Weighted WACP

Objective


TransmissionBand


After Predistortion WithOver-Weighted WACP

Objective

ffCarrier

Figure 9.4: Power amplifier output spectra, before predistortion (red), after predis-tortion with over-weighted WACP objective (black), after predistortion with under-weighted WACP objective (blue) and after ideal predistortion (green).


stage. Taking this changing sensitivity into account, we propose the use of weighting

function taper, the process of successively reducing the weighting function amplitude

WE throughout the optimization process. Compared to a fixed WE, a tapered WE

prevents the occurrence of these under and over-weighting scenarios.

While many weighting functions and associated tapers can be defined, a quadratic

weighting function (9.5) with initial WE = 100000 and 10% linear taper for each op-

timization subphase thereafter, was found to give superior performance in our work3.

This specific weighting function and taper is applicable to all DVB-T, WCDMA and

DAB target modulations.

Several considerations must be taken into account when physically computing the

WACP objective:

• Since a continuous frequency integration cannot be performed in practice, (9.3)

must be discretized with respect to frequency:

WACP = ∑LAC ∣∆f

W (f)P (f) + ∑UAC ∣∆f

W (f)P (f) (9.6)

The discrete frequency step size ∆f is chosen as a trade off between accu-

mulation speed and WACP fidelity. ∆f = B/60 was used with good effect

in the experimental analysis of this research. Being a function of modulation

bandwidth B, this step size applies to all of our target applications.

• (9.6) can be computed either via a DSP/FPGA implemented spectral esti-

mation algorithm or a software controlled spectrum analyzer. Based purely

on hardware availability, we choose the latter. With appropriate measure-

ment settings [73, 286, 287], the analyzer is instructed to sequentially sweep

its marker to each discrete frequency and report the P (f) measurement. A

subroutine then performs the final weighted accumulation.

On the laboratory transmitter testbed, this software procedure is implemented

by the function MeasureObjectiveFunction(). To be precise, this is a member

function of the object-oriented class ObjectiveFunction ACS. Corresponding

function-definition and class-declaration source code resides in project files

ObjectiveFunction ACS Templates.cpp and ObjectiveFunction ACS.h respec-

tively. Both files are located within folder Software\Cpp\ on the accompanying

DVD. Spectrum analyzer measurement settings were discussed in Section 4.5.

3Optimization schedules and subphases will be discussed further in Chapter 10


• Technically speaking, our target modulating signals (DAB, WCDMA, DVB-T)

are random processes given the inherent randomness of the input data stream.

It follows that the transmitter output signal is also a random process and

the WACP objective (9.6) is theoretically a random variable with mean and

spread.

In this respect, WACP averaging will logically provide a more reliable objec-

tive estimate and hence aid optimizer convergence. However, considering the

spectrum analyzer’s multi-second sweep time4, this averaging also leads to slow

optimizer tracking. Fortunately, by choosing robust optimization algorithms

(to be discussed in Chapter 11), the detrimental convergence effects of WACP

randomness can be mitigated and the amount of averaging reduced to zero.

Apart from a Volterra Series pruning refinement of (9.2), we have now formally

defined the optimization framework of the SPFL strategy in terms of the new WACP

objective. This framework sets the foundation for all future optimization activities.

To summarize, this chapter has discussed the fundamental aspects of predistortion

filter parameter estimation. Specifically, a new objective function called the Weighted

Adjacent Channel Power (WACP) has been proposed for the SPFL strategy. Com-

pared with traditional spot power objectives, WACP is more suited to current and

future wideband applications based on its ability to convey complete adjacent channel

behavior and furthermore discriminate between spectral distortion components.

4A function of resolution and video bandwidth

Chapter 10

Mathematical Optimization

Process

With the optimization framework now formally defined, attention is turned to the

optimization process. Despite the mathematical elegance of the optimization frame-

work, computing the optimal ho is not as simple as performing a single, one-off

optimization over the entire vector space h = [p3 ∣ p5 ∣ p7 ∣ p9 ]. In this chapter

we show that practical factors, specifically transmitter nonlinear drift and optimiza-

tion convergence reliability, demand a more complex optimization process defined in

terms of phases and schedules respectively.

10.1 Two Distinct Phases of Optimization

From an operational perspective, two distinct phases of optimization must be de-

fined, these being the Initial Setting and On-Air Adaption phases.

• The Initial Setting phase estimates the optimal predistortion filter parameters

ho at the very start of the transmitter’s operational life. This phase occurs with

the transmitter output switched to a dummy load as the regulatory spectral

mask [41] will not be met until the predistortion filter parameters approach

optimality. Once the Initial Setting phase is complete and hence the regulatory

mask is met, the transmitter is ready for broadcast and its output can be

switched to the antenna.

• In practice, a transmitter’s nonlinear transfer characteristic will drift slowly

during its operational life. This is a result of component aging (transistors

and capacitors), temperature fluctuations and power supply voltage varia-

tions [50,107,288]. This ultimately means that the predistortion filter param-

eters estimated during the Initial Setting phase do not remain optimal over

80

CHAPTER 10. MATHEMATICAL OPTIMIZATION PROCESS 81

the entire lifetime of the transmitter, hence the need for an On-Air Adaption

phase. The On-Air Adaption phase adapts the predistortion filter parameters

in order to maintain optimality whilst the transmitter’s nonlinear transfer char-

acteristic is changing. This phase must occur whilst the transmitter is on-air,

performing its intended communication function, since taking the transmitter

off-air is both undesirable and costly for the transmitter owner.

So we see that the optimization process is not a one-off event, as one might initially

assume, but rather an on-going activity performed throughout the transmitter’s

operational life.

10.2 Maximal Convergence Reliability

How we design the optimization processes for each of these Initial Setting and On-

Air Adaption phases is driven by the requirement for maximal convergence relia-

bility. Generally speaking, common reasons for poor convergence reliability are as

follows [54,74,200]:

1. An excessively large variable vector h and therefore impractically vast search space.

2. Large differences in the magnitude of variables and therefore adverse variable

sensitivity. This is commonly referred to as poor scaling.

3. A complex objective function surface exhibiting multiple inferior local minima.

4. Inability of an optimizer to hone in on a minimum once it has been located.

5. Premature exiting of an optimizer.

6. Poor estimation of Gradient vector or Hessian matrix parameters.

Armed with this knowledge, maximal convergence reliability can be achieved in

our optimization processes by employing the following respective Counter-Measures

whenever and wherever possible:

CM1. Minimize the number of variables per optimization:

A. Only include influential variables in the optimization process at all times.

B. Optimize real and imaginary parts of complex variables separately.

C. Prune the predistortion filter Baseband Volterra kernel.


CM2. Pre-scale variables with respect to nonlinear order. Experimental analysis

indicates poor scaling, with an average 40 dB reduction in variable magni-

tude per odd nonlinear order. Also avoid stepping optimizers since step size

estimation is typically unreliable when variables are poorly scaled.

CM3. Use global optimization, at least to regionalize the true global minimum and

screen out inferior local minima.

CM4. Perform precision optimization:

A. Regularly reset optimizers with fresh initialization parameters to avoid

stagnation.

B. Use a local optimizer to hone in on the minimum identified by a global

optimizer; the reason being, local optimizers tend to be more nimble

than global optimizers.

CM5. Develop a reliable means of detecting when an optimizer has achieved its

goal; whether it be to pinpoint or just regionalize a minimum.

CM6. Consider simpler but more robust gradient-free optimization algorithms as

an alternative to popular gradient based algorithms.

In essence, the requirement for maximal convergence reliability eliminates any per-

ceived weaknesses in the SPFL strategy’s parameter estimation process.

10.3 Optimization Scheduling

While running a single optimization over the entire vector space is considered the

simplest optimization process for the Initial Setting and On-Air Adaption phases, it

is also the optimization process which exhibits the minimal convergence reliability for

two reasons. Firstly, it employs the largest possible variable vector which, according

to Dot-Point 1 of the previous section, places it at higher risk of poor convergence.

Secondly, the variable vector h = [p3 ∣ p5 ∣ p7 ∣ p9 ] exhibits maximally poor scaling

which, according to Dot-Point 2 of the previous section, also places it at higher risk

of poor convergence.

It now becomes obvious that the optimization process which achieves maximal

convergence reliability is one that is based on a set of sequential optimizations over

subsets hs of the entire vector space h. This set of optimizations will from here on

be referred to as an optimization schedule.

When developing these optimization schedules, one must adhere to the ascending

nonlinear order principle first reported in Section 6.3 and alluded to thereafter. This

concept relates to the order in which predistortion filter kernels are to be computed.


Optimization Number Vector Subspace hs Objective Function

1st Optimization p3 WACP

2nd Optimization p5 WACP

3rd Optimization p7 WACP

4th Optimization p9 WACP

Table 10.1: Simplest schedule adhering to the ascending nonlinear order principle

Specifically, if parasitic growth is to be fully accounted for during the linearization

process, predistortion must be performed in ascending nonlinear order. That is, for

the specific case of optimization, kernel coefficients pm must be optimized prior to,

or in conjunction with pm+2 for m = 3, 5, 7, 9.

The simplest schedule that adheres to this principle is presented in Table 10.1.

Here, a separate optimization is allocated to each predistortion filter kernel order.

In addition to being simple, this schedule benefits from minimally sized optimization

vectors and good scaling. Based on these properties, one would consider this to be

the schedule with maximal convergence reliability. However this is not the case for

the following reason.

As discussed in Section 6.3, pm is tuned specifically to remove mth order distor-

tion. In this sense, mth order distortion must be observable at all times if the tuning

is to be successfully controlled. This observability cannot be guaranteed however

since all orders of distortion are collocated in frequency, specifically around the car-

rier, and resultant distortion is what is actually observed. Once the level ofmth order

distortion falls below that of all other orders of distortion, observability is lost and

tuning of pm fails. Based on this observability issue, we conclude that optimization

schedules cannot be based solely on the individual nonlinear order level.

The optimization schedule that we propose is based on counter-measure CM1-A

and the idea of setting hs to be the influential subset of h at all times. Here, we

define the influential subset as that which has an immediate influence on reducing

the WACP objective. That is, if m1 . . .mx are the currently dominant nonlinear

orders of distortion forming the WACP objective, then the current influential subset

is defined as [pm1∣ . . . ∣ pmx

] since pm is primarily responsible for reducing mth

order distortion. This idea is summarized in Table 10.2.


Step 1 Identify current influential subset of h

based on the WACP’s dominant distortion components

Step 2 Set hs to influential subset

Step 3 Optimize over hs

Repeat above steps until WACP objective is minimized

Table 10.2: Summarizing steps of proposed optimization schedule

By adhering to this strategy, hs will always be the minimally sized variable vector

which guarantees complete observability and for this reason, the resulting schedule

will exhibit maximal convergence reliability.

In practice, the influential subset does not need to be identified in real-time

as suggested in Table 10.2. It can in fact be predicted based on our previous

Chapter 6 & 7 analysis. In the following sections, we perform these predictions and

hence develop precise, working optimization schedules for the Initial Setting and

On-Air Adaption phases. As will be seen, the proposed schedules do indeed sat-

isfy the ascending nonlinear order principle due to the inverse relationship between

nonlinear order and distortion power.

10.4 Initial Setting Optimization Schedule

Consider the scenario at the commencement of the Initial Setting phase. Predistor-

tion is yet to be applied and therefore output distortion is purely amplifier generated.

In general, a power amplifier’s 3rd order nonlinearity will be significantly larger than

its higher order nonlinearities, as previously demonstrated in Figure 7.2 (page 64).

If follows that the WACP objective will be dominated by 3rd order distortion and

hence the influential subset can be defined as p3. According to Table 10.2, the first

scheduled optimization must therefore be performed over hs = p3. To the point,

including higher order kernels in this first optimization would have no effect other

than to jeopardize convergence.

Whilst 3rd order distortion will reduce during this initial optimization, parasitic

nonlinearities generated by the dominant 3rd order amplifier nonlinearity will cause

5th, 7th and 9th order distortion to grow significantly, as outlined in Chapter 8. It

follows that with 3rd order distortion reducing and 5th, 7th and 9th order distor-

tion growing, a converging point will be reached whereby these orders of distortion

become comparable. With a further 3– 6 dB reduction in 3rd order distortion, ob-

servability completely diminishes and the optimizer will stall, signifying the end of

the initial optimization.


The WACP objective will then be dominated by 5th, 7th and 9th order distortion

and hence the influential subset can be defined as [p5 ∣ p7 ∣ p9 ]. According to

Table 10.2, the second scheduled optimization must therefore be performed over

hs = [p5 ∣ p7 ∣ p9 ]. During this optimization, 5th, 7th and 9th order distortion will

reduce while 3rd order distortion will remain fixed, as outlined in Chapter 6. It

follows that a point will again be reached whereby these orders of distortion become

comparable. With a further 3– 6 dB combined reduction in 5th, 7th and 9th order

distortion, observability completely diminishes and the optimizer will stall, signifying

the end of the second optimization.

At this point, the WACP objective is once again dominated by 3rd order dis-

tortion and therefore the scenario is similar to the beginning of the first scheduled

optimization; just not as pronounced. This observation logically suggests that the

overall optimization schedule must be a repeating cycle of these first two scheduled

optimizations. With each cycle comes further fine tuning of the vector space es-

timate. Whilst multiple repetitions can be performed, testing indicates that only

one repetition is satisfactory for rounding out the optimization schedule. As a re-

sult, the third and fourth scheduled optimizations can be defined over hs = p3 and

hs = [p5 ∣ p7 ∣ p9 ] respectively.In terms of the WACP objective, we propose using the quadratic weighting func-

tion (9.5) with initial WE = 100000 and 10% linear taper for each scheduled opti-

mization thereafter. Of all the different functions and tapers tested with respect to

the under- and over-weighting phenomena discussed in Section 9.2, this combina-

tion stands out as being the simplest and most reliable across all of our target signal

modulations.

In accordance with counter-measure CM3 of Section 10.2, a global optimization

algorithm must be employed for the first and second scheduled optimizations in

order to locate the true global minimum of the assumed nonconvex objective. How-

ever, in accordance with counter-measure CM4-B, a more nimble local optimization

algorithm may be used for the third and fourth scheduled optimizations since the

search space will have already been narrowed to the vicinity of the global minimum

and a local refinement is all that is required. Precise algorithms to implement these

global and local optimizations will be discussed in the next chapter.

Finally, in accordance with counter-measure CM1-B, a simple but effective mea-

sure to further reduce the number of variables per optimization, hence improve

optimization convergence reliability, is to optimize each above defined subset hs

over its real and imaginary components separately. In this context, real components

are optimized first followed by imaginary components. This is because imaginary

components are generally an order of magnitude smaller than real components; as

determined during experimental analysis.


Optimization hs Optimizer WACP Weighting & Taper

1a. p3R Quadratic, WE = 105

1b. p3I Global

2a. [p5R ∣ p7R ∣ p9R ]Quadratic, WE = 104

2b. [p5I ∣ p7I ∣ p9I ]


3b. p3I Local


4b. [p5I ∣ p7I ∣ p9I ]

Table 10.3: Proposed Initial Setting optimization schedule

Based on the above analysis, the proposed optimization schedule for the Initial

Setting phase is presented in Table 10.3. Here, vector subscripts R and I signify

real and imaginary components respectively. Figure 10.1 goes on to demonstrate the

superior performance of this schedule when compared to a single global optimization

over the entire vector space h.

10.5 On-Air Adaption Optimization Schedule

As outlined in Section 10.1, the On-Air Adaption phase must adapt predistortion

filter parameters in order to maintain optimality whilst the transmitter’s nonlinear

transfer characteristic is drifting. While nonlinear drift will cause all orders of distor-

tion to grow, individual growth rates will not be the same. To be precise, growth in

3rd order distortion will be most severe due to the power amplifier’s 3rd order dom-

inance. This is demonstrated in Figure 10.2 for progressively larger forced drifts1

following the Initial Setting phase. In all cases, distortion growth is concentrated

close to transmission band edges; a clear indication of dominant 3rd order presence.

It follows that after any drift activity, the level of 3rd order distortion will rise

above that of comparable 5th, 7th and 9th order distortion and hence the optimization

scenario is similar to the beginning of the third scheduled Initial Setting optimization

discussed in the previous section. This observation logically suggests that, in terms

of vector subsets hs, the associated On-Air Adaption schedule must be a replica of

the second half of the Initial Setting phase.

1forced drift is caused by deliberate changes in amplifier supply voltage & ambient temperature

LD

Ref Lvl

12 dBm

Ref Lvl

12 dBm

RBW 30 kHz

VBW 300 kHz

SWT 2 s

RF Att 40 dB


7 dB Offset

Mixer -40 dBm

A

2RM

3RM

Unit dBm

1VIEW

2VIEW

3VIEW

1RM

*

-80

-70

-60

-50

-40

-30

-20

-10

0

-88

12

Before Predistortion

After single globaloptimization over

After proposedoptimization scheduleh

Figure 10.1: Power spectra before predistortion (red), after Initial Setting via asingle global optimization over h (magenta) and after Initial Setting via the proposedoptimization schedule (green) for a real 25Watt Class-AB push-pull power amplifierwith WCDMA signal modulation.

LD

RBW 30 kHz

VBW 300 kHz

SWT 2 s

RF Att 40 dB

Mixer -40 dBm

A

1RM

Unit dBm

2RM

4RM

1VIEW

2VIEW

3VIEW

4VIEW

3RM

Ref Lvl

12 dBm

Ref Lvl

12 dBm


7 dB Offset *

-80

-70

-60

-50

-40

-30

-20

-10

0

-88

12

After ProgressivelyLarger Forced Drifts

After Initial SettingOptimization Phase

Figure 10.2: Power spectra for progressively larger forced drifts following the InitialSetting phase. Forced drifts caused by deliberate changes in power amplifier supplyvoltage and ambient temperature. Real 25Watt Class-AB push-pull power amplifierwith WCDMA signal modulation.


The On-Air Adaption phase should commence immediately following the Initial

Setting phase and repeat indefinitely for the lifetime of the transmitter. In doing

so, the drifting objective function minimum will be under constant track and re-

convergence reliability is maximal.

As outlined in Section 10.1, nonlinear drift is attributed to component aging,

temperature fluctuations and power supply voltage variations.

• Since component aging is a continuous process, it can be assumed to cause

the objective function minimum to translate in the vector space rather than

abruptly disappear / reappear elsewhere. This translation suggests the use of

local rather than global optimization for tracking the minimum.

• Since temperature and power supply variations are centered about a mean,

they can be assumed to cause the objective function minimum to move back

and forth about the point corresponding to the mean state. This oscillation

also suggests the use of local rather than global optimization.

Precise algorithms to implement these proposed local optimizations will be discussed

in the next chapter. Irrespective of the chosen algorithms however, fine search

movements must be utilized to avoid tracking jitter during periods of little drift.

In terms of the WACP objective, we propose a quadratic weighting function with

fixed WE = 100. This is a continuation of the weighting utilized in the last scheduled

optimization of the Initial Setting phase. In practice, the lower levels of distortion

growth caused by natural nonlinear drift do not warrant weighting function taper.

Finally, as was the case for the Initial Setting phase, each defined subset hs is

optimized over its real and imaginary components separately in order to reduce the

number of variables per optimization and hence improve re-convergence reliability.

Based on the above analysis, the proposed optimization schedule for the On-Air

Adaption phase is presented in Table 10.4.

Optimization hs Optimizer WACP Weighting

1a. p3R

1b. p3I Local Quadratic, WE = 102

2a. [p5R ∣ p7R ∣ p9R ]

2b. [p5I ∣ p7I ∣ p9I ]

Repeat indefinitely for constant tracking

Table 10.4: Proposed On-Air Adaption optimization schedule


By designing the Initial Setting and On-Air Adaption schedules according to

the influential subset strategy, maximal convergence reliability is guaranteed. This

is because optimization is always restricted to the minimally sized variable vector

for which observability holds. From a high level perspective, the resulting schedules

are simply a reflection of the optimizer needing to switch between the two tasks of

removing dominant distortion (3rd order) and cleaning up those generated parasitics

(5th, 7th and 9th order).

Chapter 11

Mathematical Optimization

Algorithms

In the previous chapter, individual optimizations of the Initial Setting and On-Air

Adaption schedules were assigned local / global classifications. In this chapter, we

take the next step and assign specific optimization algorithms. With mathemati-

cal optimization being a mature field of applied mathematics, this assignment task

involves assessing the practical suitability of the many existing algorithms, in accor-

dance with the properties of our predistortion application.

11.1 Algorithm Classifications

Within the applied mathematics literature, the following algorithm classifications apply:

• Single vs Multi-Objective

• Constrained vs Unconstrained

• Discrete vs Continuous

• Gradient vs Gradient-Free

• Stochastic vs Deterministic

These classifications are in addition to the already familiar local vs global classifica-

tion. In this section, we review these classifications in the context of our optimization

framework, with the goal of understanding what type of algorithm we are actually

seeking.

90

CHAPTER 11. MATHEMATICAL OPTIMIZATION ALGORITHMS 91

Single vs Multi-Objective

If an optimization problem is defined with a single objective function B(h) then the

problem is called a single objective optimization problem. The goal is to find the

optimal vector ho which minimizes the objective.

If on the other hand an optimization problem is defined with multiple, possibly

conflicting objective functions [B1(h) B2(h) . . . Bx(h) ], then the problem is called

a multi-objective optimization problem. Given two or more of the objectives are in

general conflicting, there won’t exist a single optimal vector which simultaneously

minimizes each objective. The goal in this case is to find the optimization vector

which optimally trades off each objective according to some predefined criteria.

It is also possible to convert a multi-objective problem into a single objective

problem by weighting each objective and accumulating as follows:

B(h) =W1B1(h) +W2B2(h) + . . . +WxBx(h) (11.1)

In such cases, B(h) is specifically known as an aggregated single objective func-

tion and the optimal solution ho depends on the relative values of the specified

weights [246,266].

By definition, WACP is considered an aggregated single objective function since

it is the weighted accumulation of multiple objectives; these objectives being the

spectral powers of each adjacent channel frequency:

WACP = ∫LAC

W (f)P (f)df + ∫UAC

W (f)P (f)df (11.2)

Irrespective of whether WACP is pure or aggregated, our optimization problem is

by definition single objective and must be solved using single objective optimization

algorithms.

Constrained vs Unconstrained

Constrained optimization problems are those which have constraints placed on their

optimization vector space elements. These constraints may be simple upper / lower

bounds on individual elements or more complex inter-element linear / nonlinear,

equalities / inequalities representing physical relationships among the elements [200].

Unconstrained optimization problems on the other hand are those which don’t pos-

sess vector space element constraints. Different optimization algorithms must be

used depending on whether the problem is constrained or unconstrained.

In terms of our optimization framework, the vector space h is defined as the entire

set of predistortion filter kernel coefficients. These kernel coefficients are unbounded

and can be considered unrelated to each other in the context of optimization. In


this sense, no constraints exist on the vector space elements. It follows that our

optimization problem is by definition unconstrained and must be solved using an

unconstrained optimization algorithm.

Discrete vs Continuous

In a discrete optimization problem, elements of the optimization vector space can

only take on discrete values and therefore the vector space is considered a finite set.

This is in direct contrast to a continuous optimization problem whereby elements of

the optimization vector space can take on continuous values and therefore the opti-

mization vector space is considered an infinite set. Different optimization algorithms

are used based on whether the problem is discrete or continuous.

In terms of our optimization framework, the vector space h is defined as the entire

set of predistortion filter kernel coefficients. These kernel coefficients are continuous

by nature taking on values from the complex number plane C. In this sense, the

optimization vector space is the infinite complex set CS where S is the size of h.

It follows that our optimization problem is by definition continuous and must be

solved using a continuous optimization algorithm.

Gradient vs Gradient-Free

Gradient based optimization algorithms utilize the objective function’s first and

second order derivative characteristics (Gradient vector and Hessian matrix respec-

tively) to navigate to the local / global solution. In practice, these characteristics are

approximated via Finite Differences and Symmetric Rank 1 updating [200]. Whilst

gradient based optimization algorithms are technically superior to other forms of

optimization, they are known to be computationally intensive and susceptible to

measurement noise and vector space scaling issues.

In direct contrast, gradient-free optimization algorithms rely solely on objective

function measurements to navigate to the local / global solution. They do not require

knowledge of the objective function’s derivative characteristics. Whilst gradient-free

optimization algorithms are not as technically apt as gradient based algorithms, they

are known to be more robust and computationally efficient.

Given both gradient and gradient-free algorithms have their unique advantages,

it would be wise at this stage to consider both types as candidate algorithms.

Stochastic vs Deterministic

Stochastic optimization algorithms are those which utilize probabilistic or random

decision making. At each execution step of a stochastic algorithm, there potentially

exists more than one way to proceed. This is in direct contrast to deterministic


algorithms which possess a clear decision making process. At each execution step of

a deterministic algorithm, there exists only one way to proceed.

Deterministic algorithms are known to be thorough in their search for local and

global minima. For low and medium dimensionality problems, this is an advantage.

However for high dimensionality problems, this thoroughness leads to long run times.

In such cases, stochastic algorithms are favored as they trade in guaranteed correct-

ness of the solution for shorter run times [283].

Given both deterministic and stochastic algorithms have their unique advan-

tages, it would be wise at this stage to consider both types as candidate algorithms.

Based on the above review of algorithm classifications, we conclude that single

objective, unconstrained, continuous, local and global optimizers are required to solve

our optimization problem. These optimizers can be either gradient or gradient-free

as well as either stochastic or deterministic.

11.2 Theoretical Shortlist of Optimization Algorithms

A search of the applied mathematics literature for algorithms fitting this criteria

identifies the following theoretical shortlist of optimization algorithms:

Optimization Local or Gradient or Stochastic or

Algorithm Global Gradient-Free Deterministic

Gradient Descent L G D

Trust Region Newton L G D

Alpha Branch & Bound G G D

Nelder-Mead Simplex L GF D

Genetic G GF S

Table 11.1: Theoretical shortlist of optimization algorithms

It is noted that the Simulated Annealing algorithm (local, gradient-free, stochastic)

also fits this criteria but fails to make the shortlist. This is because the algorithm’s

central control function, the annealing schedule probability function, is unknown for

our predistortion application [141, 213]. If the annealing schedule function is set

too conservatively, the algorithm will randomly bounce around the search space

without convergence. If the annealing schedule function is set too aggressively, the


algorithm will revert to random steps in the direction of objective function reduction

but with unknown convergence rate. Estimating the annealing schedule probability

function requires experimental trial and error and hence the algorithm is considered

too unreliable in the context of achieving maximal convergence reliability.

A complete treatise of Table 11.1’s shortlisted algorithms is presented in Ap-

pendix A. Readers are strongly encouraged to consult this Appendix in order to

familiarize themselves with the inner workings of each algorithm.

11.3 Practical Suitability of Shortlisted Algorithms

Leveraging on the theory and hardware implementation of Appendix A, we now pro-

ceed to examine the practical suitability of each shortlisted algorithm in accordance

with the properties of our predistortion application. We start with local algorithms

then proceed globally.

Gradient Descent (Local)

� The deterministic nature and hence clear decision making process of this al-

gorithm is favorable.

� For even-numbered local optimizations of the Initial Setting and On-Air Adap-

tion schedules, where hs spans multiple orders (5th ∣7th ∣9th), poor scaling will

lead to step size sensitivity. As discussed in Section 10.2, experimental analysis

indicates an average 40 dB reduction in variable magnitude per odd nonlinear

order. In this sense, if step size is chosen to match the larger magnitude of

lower order variables, higher order variables will become extremely sensitive

causing dramatic changes in the objective function. If step size is chosen to

match the smaller magnitude of higher order variables, little progress is made

in optimizing the lower order variables. In practice it is very difficult to find

an adequate tradeoff, even when employing variable pre-scaling.

� For odd-numbered local optimizations of the Initial Setting and On-Air Adap-

tion schedules, where hs is restricted to 3rd order variables and scaling is

good, estimation of the Gradient vector and hence step direction will be sensi-

tive to WACP randomness. As discussed in Section 9.2, the WACP objective

is theoretically a random variable with mean and spread. In theory, such a sen-

sitivity problem could be overcome by averaging WACP measurements during

the Gradient estimation process. However with:

● at least two seconds per raw WACP measurement

↪ determined by spectrum analyzer sweep, resolution/video BW


● at least 5 raw WACP measurements per averaged WACP measurement

↪ to adequately reduce variance

● at least 3n averaged WACP measurements per Gradient vector estimate1

↪ determined by the robust least squares technique of Appendix B.1

this averaging would lead to a very slow stepping / convergence rate and hence

only be viable for small variable vectors.

Since we generally can’t rely on the prospect of small variable vectors, especially

for future wider band modulation strategies, this algorithm cannot be used for either

even- or odd-numbered scheduled optimizations and hence is considered unsuitable

for our predistortion application.

Trust Region Newton (Local)



� This algorithm utilizes both first order (Gradient vector) and second order

(Hessian matrix) derivative characteristics to model the objective function

locally at each step. The domain of this model is further verified by an ap-

propriate trust region. This high level of objective conformity, coupled with

analytic computation of the model minimum, ensures poor scaling is not a

problem.

� At each step, estimation of the Gradient vector /Hessian matrix and hence

quadratic objective model will be sensitive to WACP randomness. As dis-

cussed in Section 9.2, the WACP objective is theoretically a random variable

with mean and spread. This sensitivity will subsequently diminish the associ-

ated trust region and hence modeling domain, leading to reduced convergence

rate and reliability. In theory, such a problem could be overcome by averag-

ing WACP measurements during the Gradient /Hessian estimation processes.

However, with:





1n being the dimension of the hs vector


● at least 3n averaged WACP measurements per Gradient vector estimate

↪ determined by the robust least squares technique of Appendix B.1

● at least 3n2+3n averaged WACP measurements per Hessian matrix estimate

↪ determined by the robust hybridized technique of Appendix B.2

this averaging would lead to an excruciatingly slow stepping / convergence rate,

worse than Gradient Descent due to additional Hessian and trust region esti-

mation, and hence be practically unviable even for small variable vectors. This

same conclusion is reached for Symmetric-Rank-1 Updating of the Hessian es-

timate.

Despite being resilient to poor scaling, this last � renders the algorithm unsuitable


Nelder-Mead Simplex (Local)



� Since all simplex geometric manoeuvres (reflection, contraction, expansion,

shrinkage) are dimensionally orthogonal, pre-scaling variables with respect to

nonlinear order during simplex initialization allows the algorithm to accom-

modate for poor scaling. This is in accordance with counter-measure CM2 of

Section 10.2. In practice, pre-scaling magnitude for each nonlinear order is set

to be the variable increment size which induces a small but observable change

in the output distortion spectrum. In accordance with CM4, routine resetting

of the simplex also ensures scaling induced stagnation is avoided.

� WACP randomness will only interfere with this objective measurement and

comparison strategy when simplex vertices are all of the same or similar ob-

jective value. Generally speaking, this only occurs once the simplex has ge-

ometrically contracted within the desired local minimum and hence achieved

its goal. It follows that this gradient-free algorithm can cope with WACP ran-

domness without having to rely on WACP measurement averaging and is thus

capable of large-scale optimization.

With its resilience to poor scaling and WACP randomness, this algorithm is consid-

ered highly robust and therefore well suited to our predistortion application.


Alpha Branch & Bound (Global)



� For each boxed region, poor variable scaling will be transfered from the objec-

tive function to the convex under-estimating function L(h). This algorithm’s

ability to cope with poor variable scaling is thus determined by the local op-

timizer chosen to estimate each region’s single L(h) minimum. In addition to

poor scaling, this bounding local optimizer will experience inherited WACP

randomness. As discussed above, the Nelder-Mead Simplex algorithm is re-

silient to this poor scaling and objective randomness, and is thus the perfect

choice of local optimizer to ensure the ABB algorithm performs well in the

presence of poor variable scaling.

� For each boxed region, spot Hessian matrix estimates will be sensitive to

WACP randomness. This sensitivity will subsequently diminish the Hessian

Interval matrix estimate and therefore convex under-estimating function L(h),leading to either over-bounding (slow convergence) or under-bounding (no con-

vergence). In theory, such a problem could be overcome by averaging WACP

measurements during the spot Hessian estimation process. However, with:





● at least 3n2+3n averaged WACP measurements per Hessian matrix estimate

↪ determined by the robust hybridized technique of Appendix B.2

● at least 10 spot Hessian matrix estimates per Hessian Interval matrix / region

↪ to reduce sampling bias

● many regions initialized / branched over the lifetime of the algorithm

this averaging would lead to an excruciatingly slow bounding / convergence

rate and hence be practically unviable even for small variable vectors.

Despite being resilient to poor scaling, this last � renders the algorithm unsuitable



Genetic (Global)

� Global convergence is unaffected by the stochastic nature of this algorithm

provided the following measures are taken:

1. An initially large number of chromosomes χ are presented for selection

into the population pool.

2. Fresh chromosomes are introduced into the mating pool of each evolu-

tionary refinement cycle.

3. Evolutionary refinement continues until a small number ϑ of distinct chro-

mosome concentrations form.

Measures 1 and 2 ensure appropriate seeding whilst measure 3 ensures non-

premature exiting of the algorithm in accordance with CM5 of Section 10.2.

� Since parent genes are treated independently during reproduction, pre-scaling

initial gene bounds and mutation variance with respect to nonlinear order

allows the algorithm to accommodate poor scaling. This is in accordance with

counter-measure CM2 of Section 10.2. As was the case for the Nelder-Mead

Simplex algorithm, pre-scaling magnitude for each nonlinear order is set to be

the variable increment size which induces a small but observable change in the

output distortion spectrum.

� WACP randomness can be envisioned as chromosome fitness jitter. In the

context of selection and reproduction, chromosome fitness jitter will introduce

parent gene mutation and hence increase the effective variance of offspring

mutation. By proactively reducing the standard Normal variance of offspring

mutation by 5%, this additional parent mutation can be successfully accom-

modated. In the context of the population pool and candidate solution space,

chromosome fitness jitter will mutate chromosome concentration regions. Since

mutation is unbiased however, these concentration regions remain centered on

candidate minima and the algorithm proceeds as normal. In both cases above,

this gradient-free algorithm is seen to cope with WACP randomness without

having to rely on WACP measurement averaging and is thus capable of large-

scale optimization.

With its resilience to poor scaling and WACP randomness, this algorithm is consid-

ered highly robust and therefore well suited to our predistortion application.


11.4 Selected Optimization Algorithms

We conclude from this overall analysis that the local Nelder-Mead Simplex and

global Genetic algorithms are the most suitable shortlisted algorithms for our pre-

distortion application. Both of these algorithms are robust, gradient-free, objective

measurement and comparison strategies. Their choice is thus in accordance with

CM6 of Section 10.2. Whilst all other shortlisted algorithms may have far superior

search strategies (being gradient-based), their downfall is the excessive time spent

estimating Gradient and Hessian characteristics of the WACP objective function.

Internal Parameter Settings

As discussed in Sections A.4 and A.5, the Nelder-Mead Simplex and Genetic al-

gorithms possess several internal parameter settings. While hardware testing has

revealed that algorithm performance is not overly sensitive to common sense pa-

rameter settings, the values outlined below are recommended as an initial guide.

For the Nelder-Mead Simplex algorithm (summarized in Figure A.6), three pa-

rameters require setting. The first is the simplex reset rate which controls stagnation

and hence rate of convergence. A 40 iteration reset rate is used successfully in our

performance testing. The second parameter that needs setting is d, the starting

point increment of the initial simplex. As stated in Section A.4, this just needs to

be set small compared to the dimension of the search space. We use a proportional

value of 10% in our performance testing. The final parameter that needs setting

is the exit criterion Υ0. Once again, this just needs to be set small and a value of

Υ0 = 10−2 has proven very reliable in our testing.

For the Genetic algorithm (summarized in Figure A.10), five parameters exist;

although only two of these are considered primary, requiring upfront setting. The

remaining three parameters, considered secondary, are derived accordingly. The first

of the primary parameters is the working Population Pool size ξ, which we set to 50

chromosomes in our performance testing. The second of the primary parameters is

the number of chromosome concentration regions ϑ at which evolutionary refinement

is halted. As stated in Section A.5, this just needs to be set small and we find a

value of ϑ = 4 is very reliable.

The three secondary parameters are the Mating Pool size κ, Offspring Pool

size � and initial Population Pool size χ. These are logically derived from the

primary working Population Pool size ξ, to implement the Selection, Reproduction

and Seeding functions respectively. With our earlier setting of ξ = 50, values of

κ = 10, � = 50 and χ = 200 are derived accordingly.


11.5 Refined Optimization Schedules

With the Nelder-Mead Simplex and Genetic algorithms now determined to be the

local and global algorithms of choice for our predistortion application, Tables 11.2

and 11.3, with refined third column optimizers, hereby represent the finalized Initial

Setting and On-Air Adaption schedules.



1b. p3I Genetic


2b. [p5I ∣ p7I ∣ p9I ]


3b. p3I Nelder-Mead

4a. [p5R ∣ p7R ∣ p9R ] SimplexQuadratic, WE = 102

4b. [p5I ∣ p7I ∣ p9I ]

Table 11.2: Finalized Initial Setting optimization schedule


1a. p3R

1b. p3I Nelder-Mead Quadratic, WE = 102

2a. [p5R ∣ p7R ∣ p9R ] Simplex

2b. [p5I ∣ p7I ∣ p9I ]


Table 11.3: Finalized On-Air Adaption optimization schedule

Chapter 12

Pruning The Predistorter

Volterra Kernel

In this chapter, we cover the final implementation aspects of the predistortion filter

architecture, specifically Volterra Series pruning and associated memory estimation.

This subsequently allows us to refine the optimization vector space h and hence gain

a practical understanding of optimization loading.

Currently, our predistortion filter is modeled by the Baseband Volterra Series

with no linear distortion and maximum 9th order nonlinearity. This model was

derived chronologically as follows:

1. In Section 5.2, the power amplifier was modeled as the pure Volterra Series

in the RF transmitter model. Upon converting this RF transmitter model to

its baseband equivalent, the power amplifier was found to take on the variant

Baseband Volterra Series architecture characterized by odd ordered, complex

kernels and conjugated product terms.

2. In Section 5.3, it was postulated that the predistortion filter with the greatest

potential for canceling the nonlinear memory effects of the amplifier would be

one that shares the same general architecture as the amplifier. Based on this

thinking, the predistortion filter was modeled as the discrete-time Baseband

Volterra Series.

3. In Section 6.2, the predistortion filter’s first order operator was set to be trans-

parent (unit impulse kernel) since digital predistortion, as standard practice,

does not attempt to compensate for an amplifier’s linear distortion.

4. In Chapter 8, a trade off analysis between linearization performance and com-

putational complexity saw the the maximum order of predistorter nonlinearity

chosen to be 9.

101

CHAPTER 12. PRUNING THE PREDISTORTER VOLTERRA KERNEL 102

In line with the original operator notation of Chapter 5, the predistortion filter

architecture is thus represented as:

P [ s[k] ] =∞∑

odd m=1Pm[ s[k] ]

where

Pm[ s[k] ] =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

s[k] for m = 1

∞∑i1=0

⋯∞∑im=0

pm[i1, . . . , im] s[k − i1]⋯s[k − i ⌈m2⌉] s∗[k − i ⌈m

2⌉+1]⋯s∗[k − im]

for 3 ≤ m ≤ 9

0 for m ≥ 11

In this chapter, we choose a more compact representation for this predistortion fil-

ter, derived by absorbing all three mth order conditions into a single equation and

grouping unconjugated / conjugated input product terms:

P [ s[k] ] = s[k] +4

∑a=1

⎡⎢⎢⎢⎢⎣

∞∑i1=0

⋯∞∑

i2a+1=0

⎛⎝p2a+1[i1, . . . , i2a+1]

a+1∏x=1

s[k−ix]2a+1∏

y=a+2s∗[k−iy]

⎞⎠

⎤⎥⎥⎥⎥⎦(12.1)

As discussed in Chapter 5, the large kernel of this generally applicable architecture

requires pruning prior to implementation. Pruning is the process of stripping away

predicted non-dominant kernel coefficients. Since the amplifier memory effects we

are attempting to cancel are specific in origin and decline with time [299], most ker-

nel coefficients of (12.1) will actually be non-dominant and hence have little effect

in the linearization process. Pruning will thus dramatically reduce the size of our

predistortion filter kernel and hence optimization vector space h, without adversely

affecting linearization potential. In accordance with counter-measure CM1-C of Sec-

tion 10.2, a smaller optimization vector space ultimately increases our optimization

convergence reliability.

12.1 The Pruning Strategy

The pruning strategy we ultimately derive must have origins within the power am-

plifier modeling community. This is in accordance with our original postulation that

the predistortion filter with the greatest potential for canceling the nonlinear memory

effects of the amplifier, is one that shares the same architecture as the amplifier.


The power amplifier modeling community propose several pruning strategies includ-

ing Physical Knowledge, Near Diagonality Restriction, Dynamic Deviation Reduction

and Volterra Behavioral Wideband. As outlined below, each of these strategies use

different mechanisms for predicting the dominant kernel coefficients. For a complete

treatise of each strategy, the reader is directed to the references provided:

Physical Knowledge [305] - In this strategy, physical behaviors of a broad range

of real amplifiers are summarized and abstracted to form a functional block

model possessing feedback. Coefficients of this model are then regrouped and

generalized to form equivalent Volterra kernels.

Near Diagonality Restriction [299, 306] - This strategy is an extension of the

well known memory polynomial model [138, 147] with near-diagonal memory

terms being retained in addition to those residing on the diagonal. Near-

diagonal terms are restricted to offset columns of the predefined V-Vector

input structure whose dimensions are determined by memory.

Dynamic Deviation Reduction [302, 304] - In this strategy, static nonlinear-

ities and low order dynamics are identified to be the dominant sources of

distortion in real power amplifiers. Kernel coefficients are subsequently differ-

entiated according to their dynamic order and retained only if low ordered.

Volterra Behavioral Wideband [45] - Under the assumption of wideband oper-

ation, this strategy identifies dominant frequency-domain interactions existing

between dependent control sources and ports of a general nonlinear network.

These interactions, represented by multi-dimensional nonlinear transfer func-

tions, are subsequently transformed to the time-domain via Fourier Inverses,

to obtain the dominant Volterra kernel coefficients.

Of all these pruning strategies, Volterra Behavioral Wideband is considered the most

suitable for wideband applications based on its operational bandwidth assumptions

and frequency-domain development. For this reason, it is chosen as our basis strategy

and applied to (12.1) to give:

P [ s[k] ] = s[k] +4

∑a=1

⎡⎢⎢⎢⎢⎣

M

∑i1=0

⋯M

∑ia=0

⎛⎝p2a+1[i1, . . . , ia]s[k]

a

∏x=1

∣s[k − ix]∣2⎞⎠

⎤⎥⎥⎥⎥⎦(12.2)

It can be seen from (12.2) that the 3rd order kernel (a = 1) now grows linearly

with respect to finite memory length M and is hence a practically manageable size.

However, the same cannot be said for higher order kernels which continue to grow

exponentially. At this point we have three options:


1. Stick with this wideband pruning strategy, accepting the risk that optimization

convergence reliability could be jeopardized by high order kernel size.

2. Choose a more aggressive narrower-band pruning strategy but jeopardize over-

all linearization performance due to lack of model fidelity.

3. Apply additional higher order pruning.

Since the first two options aren’t suitable from a performance perspective, we are

forced in the direction of additional higher order pruning and hence an overall

hybridized pruning strategy.

For this additional higher order pruning, we turn to the Dynamic Deviation

Reduction strategy and its fundamental principle that high dynamic order terms

contribute very little to model fidelity and hence can be removed to reduce kernel

size. The dynamic order of a term in the baseband Volterra Series refers to the

number of delayed inputs forming that term. This should not be confused with the

nonlinear order of a term which refers to the total number of inputs forming that

term. For the Volterra Behavioral Wideband pruning of (12.2), the dynamic order

of a (2a+ 1)th nonlinear order term ranges from 0 to 2a. For example the arbitrary

term:

p7[3,2,5] s[k] ∣s[k − 3]∣2 ∣s[k − 2]∣2 ∣s[k − 5]∣2 (12.3)

has a nonlinear order of 7 (a = 3) and a dynamic order of 6, whilst the arbitrary

term:

p7[0,2,5] s[k] ∣s[k]∣2 ∣s[k − 2]∣2 ∣s[k − 5]∣2 (12.4)

has a nonlinear order of 7 (a = 3) but a dynamic order of 4.

Since the 3rd order kernel of (12.2) comprises terms up to 2nd dynamic order, it is

reasonable to restrict higher order kernels to 2nd dynamic order also. Applying this

Dynamic Deviation Reduction to (12.2) then gives:

P [ s[k] ] = s[k] +4

∑a=1

⎡⎢⎢⎢⎢⎣

M

∑i=0

p2a+1[i]s[k] ∣s[k]∣2(a−1) ∣s[k − i]∣2

⎤⎥⎥⎥⎥⎦(12.5)

It can be seen from (12.5) that all kernels now grow linearly with respect to memory

length M and are hence a practically manageable size. While this desirable outcome

marks the end of pruning from the perspective of electrical amplifier matching, one

further implementation related pruning opportunity exists as follows.

The input signal to the digital predistortion filter must be oversampled (Inphase

and Quadrature) by at least the highest predistortion filter nonlinearity in order to

prevent spectral regrowth aliasing. This heavy oversampling leads to an input signal


with a very narrow discrete-time spectral bandwidth Bd = Bc/Fs. Here, Bc is the

signal’s continuous-time spectral bandwidth and Fs is the sampling frequency. As

a result, the change between adjacent input signal samples can be considered very

small to the point where groups of R input samples (R being small) can be assumed

equal. With this assumption, (12.5) can be refined with R-sample delay increments

instead of single-sample delay increments without loss in fidelity as follows:

P [ s[k] ] = s[k] +4

∑a=1

⎡⎢⎢⎢⎢⎣

⌈M+1R⌉−1

∑i=0

p2a+1[i]s[k] ∣s[k]∣2(a−1) ∣s[k −Ri]∣2

⎤⎥⎥⎥⎥⎦(12.6)

In this refinement, ⌈ ⋅ ⌉ represents the ceiling operator and˜denotes kernel transfor-

mation. The precise form of this transformation is considered irrelevant however

since (12.6) kernel estimation is performed independently of (12.5).

These larger R-sample delay increments not only lead to a further (approximate)

R-fold pruning of the predistortion filter kernel but also make the architecture inde-

pendent of hardware sample rate implementation. The value of R must be estimated

from the input signal’s discrete-time bandwidth with a smaller discrete-time band-

width allowing a greater value of R and vice versa. Discrete-time bandwidths and

corresponding values of R, as used in the experimental analysis of this research, are

presented in Table 12.1.

Signal Bc Fs Bd =Bc/Fs R

Modulation (MHz) (MHz) (cycles / sample) (samples)

DAB 1.537 65.536 0.023 5

WCDMA 4.096 92.160 0.044 3

DVB-T 6.656 64.000 0.104 2

Table 12.1: R-sample delay increments

It is noted here that the sampling rate of the DVB-T signal is lower than that of the

DAB and WCDMA signals despite exhibiting a greater continuous-time bandwidth.

This is because the next power of two IFFT would bring the sampling rate to greater

than what the implementation hardware is capable. The current sampling rate is

satisfactory however given it oversamples the DVB-T signal by at least the highest

predistortion filter nonlinearity and therefore avoids spectral regrowth aliasing.

In all cases, a conservative estimate of R is advisable. Over-estimating R in


order to gain extra kernel pruning will only invalidate the underlying assumption

that groups of R input samples are equal and hence lead to degraded linearization

performance. It is also noted that although the predistortion filter has been re-

fined to operate with internal R-sample delay increments, it is still clocked at the

oversampling rate to avoid spectral regrowth aliasing.

The novel, triple pruned baseband Volterra Series (12.6) represents the final dig-

ital predistortion filter architecture. Quite simply, any further pruning will remove

dominant kernel coefficients and hence jeopardize linearization performance. This

predistortion filter architecture is implemented in software on the laboratory trans-

mitter testbed, specifically via the object class Predistorter Volterra and its member

function filter(). Corresponding class-declaration and function-definition source code

resides in project files Predistorter Volterra.h and Predistorter Volterra Templates.cpp

respectively. Both files are located within folder Software\Cpp\ on the accompany-

ing DVD.

Based on its derivation, we call the novel hybridized pruning strategy Volterra

Behavioral Wideband with Reduced Dynamic Order and R-Sample Delay Increments.

For clarity, this final digital predistortion filter architecture is expanded in (12.7) in

terms of its individual nonlinear operators. In this representation, kernel accentua-

tion is dropped for the sake of notation simplicity:

P [ s[k] ] = s[k]

+⌈M+1

R⌉−1

∑i=0

p3[i] s[k] ∣s[k −Ri]∣2

+⌈M+1

R⌉−1

∑i=0

p5[i] s[k] ∣s[k]∣2 ∣s[k −Ri]∣2

+⌈M+1

R⌉−1

∑i=0

p7[i] s[k] ∣s[k]∣4 ∣s[k −Ri]∣2

+⌈M+1

R⌉−1

∑i=0

p9[i] s[k] ∣s[k]∣6 ∣s[k −Ri]∣2

(12.7)

In the next section, we set out to estimate the predistortion filter’s finite memory

length M ; the final variable in (12.7) yet to be quantified.

12.2 Memory Estimation

The most intuitive approach to estimating predistortion filter memory is to first

estimate transmitter memory, via test signal injection and vector network analy-

sis / correlation [166,202,279], and then translate this estimate back across to the pre-

distortion filter. As signal modulation bandwidth grows however, this system identi-


fication approach becomes increasingly inaccurate, specifically under-estimating [80,

239, 272]. Hence for applications such as DAB, WCDMA and DVB-T, we propose

a more reliable experimental procedure for estimating predistortion filter memory.

This procedure is based on probing the transmitter with memory specific distortion

created by the predistortion filter and observing its effect on the signature of the

output adjacent channel power spectrum. Signature here refers to the shape of the

spectrum. The idea is simple; if a memory specific probing shows potential in re-

ducing spectral power evenly across the adjacent channel band, then that memory

component is considered present.

It is important to understand that in this procedure, the predistortion filter is

kept in place but isn’t operated in the normal sense to linearize the transmitter.

Rather, it is operated specifically to probe the transmitter with memory specific

distortion. This involves:

1. setting its internal R-sample delay increment to unity. As will become evident

later, this allows a finer memory estimate to be achieved.

2. setting a single 3rd order kernel coefficient p3[i] to a nonzero value whilst

keeping all other kernel coefficients zero. Here, the choice of i ultimately

determines the memory associated with the probing distortion. In this respect,

the predistortion filter reduces to:

P [ s[k] ] = s[k] + p3[i]s[k] ∣s[k − i]∣2 (12.8)

The nonzero 3rd order kernel coefficient p3[i] can be chosen as real or complex

although experience has shown real to be sufficient. Its sign and magnitude

are chosen with the intent of reducing the adjacent channel power spectrum

by an observable amount (≈2dB).

Since system memory is responsible for frequency dependent behavior [52,53], prob-

ing the transmitter with memory specific distortion i will have the following quali-

tative effects:

• When i is set less than the required memory, the probing distortion can be

expected to add destructively (assuming correct sign of p3[i]) with existing

transmitter distortion and the resulting power spectrum will generally reduce

evenly across the adjacent channel band. In this sense, the signature of the

adjacent channel power spectrum does not change.

• When i is set greater than the required memory, the probing distortion can be

expected to have no memory matched transmitter distortion and as a result

the predistortion filter will effectively introduce distortion which has a different


adjacent channel spectral signature to any distortion which currently exists.

It follows that the resulting adjacent channel power spectrum will not only

change magnitude, but more importantly change its spectral signature.

It logically follows that if a sweep of i from unity upwards is performed, a change

in the signature of the adjacent channel power spectrum will be observed when i

reaches the required memory. It is this signature changing value of i which we use

to estimate M .

This concept of sweep-probing the transmitter with memory specific distortion

and looking for changes in the signature of the adjacent channel power spectrum

is demonstrated in Figures 12.1 / 12.2 for the laboratory transmitter testbed and

WCDMA/DVB-T signal modulation. In the following analysis, we focus on the

WCDMA case alone, purely to reduce the amount of subfigure referencing, however

the exact same conclusions can be drawn from the DVB-T case.

Subfigure 12.1a is representative of the output power spectrum after probing

with 0 ≤ i ≤ 15, Subfigure 12.1b is representative of the output power spectrum after

probing with 16 ≤ i ≤ 20 and Subfigure 12.1c is representative of the output power

spectrum after probing with 21 ≤ i ≤ 30. It is noted that ranges of i are presented

here, instead of individual values, to limit the number of plots.

It is observed that for 0 ≤ i ≤ 15 (Subfigure 12.1a), the signature of the probed

spectrum remains similar to that of the unprobed spectrum, remembering that sig-

nature here refers to the shape of the spectrum, not its magnitude. The observed

change in magnitude is due to the probing distortion adding destructively with ex-

isting transmitter distortion.

As i increases however, in this case for 16 ≤ i ≤ 20 (Subfigure 12.1b), the signa-

ture of the probed spectrum begins to change with a slight ripple forming across

the adjacent channel. This range of i marks the point when the probing distortion

runs out of memory matched transmitter distortion and therefore the probing pre-

distortion filter is effectively introducing distortion which has a different spectral

signature to any transmitter distortion which currently exists. As demonstrated

here in practice, the change in spectral signature is a continuous process occurring

over several samples, in this case 5 (16 ≤ i ≤ 20). In this sense, ambiguity can arise in

terms of whether to choose the first or last of these samples to represent predistor-

tion filter memory M . However, it is always advisable to over-estimate rather than

under-estimate memory for two reasons. Firstly, under-estimation is another form

of Volterra Series pruning which leads to degraded linearization performance and

secondly, as discussed in Section 9.2, the weighting of the WACP objective provides

added robustness against redundant memory components. Based on this reasoning,

predistortion filter memory is estimated here as M = 20.

LD

Ref Lvl

12 dBm

Ref Lvl

12 dBm

RBW 30 kHz

VBW 300 kHz

SWT 2 s

RF Att 40 dB


7 dB Offset

Mixer -40 dBm

A

2RM

Unit dBm

2VIEW

3VIEW 3RM

*

-80

-70

-60

-50

-40

-30

-20

-10

0

-88

12

withoutprobing

withprobing

0 ≤ i ≤ 15

(a)

LD

Ref Lvl

12 dBm

Ref Lvl

12 dBm

RBW 30 kHz

VBW 300 kHz

SWT 2 s

RF Att 40 dB


7 dB Offset

Mixer -40 dBm

A

2RM

Unit dBm

3RM

2VIEW

3VIEW

*

-80

-70

-60

-50

-40

-30

-20

-10

0

-88

12

withoutprobing

withprobing

16 ≤ i ≤ 20

(b)

LD

Ref Lvl

12 dBm

Ref Lvl

12 dBm

RBW 30 kHz

VBW 300 kHz

SWT 2 s

RF Att 40 dB


7 dB Offset

Mixer -40 dBm

A

2RM

Unit dBm

3RM

2VIEW

3VIEW

*

-80

-70

-60

-50

-40

-30

-20

-10

0

-88

12

withoutprobing

withprobing

21 ≤ i ≤ 30

(c)

Figure 12.1: Output power spectrum of transmitter testbed(WCDMA signal modulation) with probing representative of(a) 0 ≤ i ≤ 15 (b) 16 ≤ i ≤ 20 (c) 21 ≤ i ≤ 30

LD

Ref Lvl

14.5 dBm

Ref Lvl

14.5 dBm

RBW 30 kHz

VBW 300 kHz

SWT 2 s

RF Att 40 dB


9.5 dB Offset

Mixer -40 dBm

A

2RM

3RM

Unit dBm

2VIEW

3VIEW

*

-80

-70

-60

-50

-40

-30

-20

-10

0

10

-85.5

14.5

withoutprobing

withprobing

0 ≤ i ≤ 11

(a)

LD

Ref Lvl

14.5 dBm

Ref Lvl

14.5 dBm

RBW 30 kHz

VBW 300 kHz

SWT 2 s

RF Att 40 dB


9.5 dB Offset

Mixer -40 dBm

A

2RM

Unit dBm

3RM

2VIEW

3VIEW

*

-80

-70

-60

-50

-40

-30

-20

-10

0

10

-85.5

14.5

withoutprobing

withprobing

12 ≤ i ≤ 14

(b)

LD

Ref Lvl

14.5 dBm

Ref Lvl

14.5 dBm

RBW 30 kHz

VBW 300 kHz

SWT 2 s

RF Att 40 dB


9.5 dB Offset

Mixer -40 dBm

A

2RM

Unit dBm

3RM

2VIEW

3VIEW

*

-80

-70

-60

-50

-40

-30

-20

-10

0

10

-85.5

14.5

withoutprobing

withprobing

15 ≤ i ≤ 20

(c)

Figure 12.2: Output power spectrum of transmitter testbed(DVB-T signal modulation) with probing representative of(a) 0 ≤ i ≤ 11 (b) 12 ≤ i ≤ 14 (c) 15 ≤ i ≤ 20


For i > M , in this case 21 ≤ i ≤ 30 (Subfigure 12.1c), the ripple in the signature

of the probed spectrum is observed to compress and slide up towards the transmit

channel, hence making room for an additional ripple in the outer adjacent channel

region (compared to Subfigure 12.1b). This pronounced rippling effect, and hence

frequency dependent behavior, continues for i > 30.

Since predistortion filter memory M is assumed independent of nonlinear order

in (12.7), we are in fact free to probe the transmitter with any order of nonlinear

distortion via (12.8). 3rd order is specifically chosen however since 3rd order trans-

mitter nonlinearity is dominant and hence its power spectrum, and any change in its

power spectrum caused by the probing predistortion filter, is completely observable.

This is not the case for higher orders.

Compared to the traditional approach of estimating transmitter memory and

then translating this estimate back to predistortion filter memory, the proposed ex-

perimental approach is considered simpler because it avoids the need for specific

test signal injection and vector network analysis / correlation. It is also considered

more direct and accurate because it estimates memory directly with the predistor-

tion filter in place, thus matching the predistortion filter architecture used. That

is, the estimate is based on exactly what the predistortion filter and transmitter

would experience in practice during optimization, thus resulting in a more accurate

estimate.

Table 12.2 presents the memory estimates M obtained when this procedure is

performed on the transmitter testbed for all target signal modulations. Here, Ts rep-

resents discrete-time sampling period whilst Mc represents continuous-time equiva-

lent memory. As can be seen from the last column, Mc is virtually constant across

the different signal modulations, thus providing confirmation of the estimates M .

These memory estimates are also consistent with independent general estimates pre-

sented in the literature [44,45,296,303].

Signal Ts M Mc =MTs

Modulation (nsec) (samples) (nsec)

DAB 15.259 14 213.623

WCDMA 10.851 20 217.014

DVB-T 15.625 14 218.750

Table 12.2: Predistortion filter memory estimates


12.3 Refined Optimization Vector Space

The optimization vector space h was initially defined in Section 9.1 in terms of the

predistortion filter’s unpruned Volterra kernel. Specific reference is made to (9.1)

and (9.2) on page 73. With the predistortion filter now pruned according to (12.7)

however, this definition can be refined as:

h = [p3 ∣ p5 ∣ p7 ∣ p9 ] (12.9)

where for m = 3, 5, 7, 9 :

pm =⎧⎪⎪⎨⎪⎪⎩pm[i] for i = 0 → ⌈M + 1

R⌉ − 1

⎫⎪⎪⎬⎪⎪⎭(12.10)

Here, pm[i] represents the pruned mth order Volterra kernel of (12.7) and ⌈ ⋅ ⌉ repre-sents the ceiling operator. With the vector space now finite in dimension, its overall

size S can be formulated as:

S = 4 ⌈M + 1

R⌉ complex variables (12.11)

Directly related to S, though slightly more insightful in terms of optimization con-

vergence reliability, the maximum optimization load L can also be formulated as:

L = 3 ⌈M + 1

R⌉ real variables (12.12)

L represents the size of the largest optimization subsets defined within the Initial

Setting and On-Air Adaption schedules (page 100), specifically [p5R ∣ p7R ∣ p9R ]and [p5I ∣ p7I ∣ p9I ]. Expectedly, L grows with predistorter memory length M and

decreases with the coarseness of the R-sample delay increment. With the values

of R and M already quantified in Tables 12.1 and 12.2 respectively, L for each

target application can be quantified via simple (12.12) substitution, as summarized

in Table 12.3.


Signal Bc R M L

Modulation (MHz) (samples) (samples) (real variables)

DAB 1.537 5 14 9

WCDMA 4.096 3 20 21

DVB-T 6.656 2 14 24

Table 12.3: Maximum optimization load L for each target application

It can be seen from this table that maximum optimization load L increases with sig-

nal modulation bandwidth Bc, reaching a maximum of 24 real variables for DVB-T.

Providing Section 11.3 and 11.4 guidelines are adhered to, this figure is easily accom-

modated by the robust Nelder-Mead Simplex and Genetic algorithms and therefore

any concerns regarding optimizer loading and convergence reliability can be laid to

rest.

Before leaving this chapter, it is important to reiterate the role of pruning in the

context of the entire algorithm. As initially outlined in the introduction of Chap-

ter 5, we chose from the very outset to develop this predistortion algorithm around

an unpruned Volterra Series model in order to establish generality and widespread

applicability. Only at the very end would we apply pruning to match the algorithm

to the target modulation standard. Based on this design principle, the pruning strat-

egy can be considered independent of the core underlying predistortion algorithm and

hence completely interchangeable if the need ever arises in the future. For instance, if

an alternative (yet to be developed) pruning strategy is identified to be more suited to

the power amplifier hardware or modulation standards of the future, it is completely

feasible to replace the hybrid pruning strategy of this chapter without affecting the

operation of the core predistortion algorithm. This front-end interchangeability leads

to an algorithm with an indefinite application life.

As alluded to above, this chapter on predistortion filter kernel pruning marks the

end of all algorithm development. In the following chapters we turn attention to

algorithm performance analysis.

Chapter 13

Performance Baseline

In this chapter, we review the quantitative performance of current wideband predis-

tortion systems in order to derive a performance baseline for our technique. This

baseline, to be used in the next chapter’s performance testing, will represent the

minimum level of performance our technique must achieve if it is to be considered

state-of-the-art.

Predistortion performance is specifically quantified in the frequency-domain by

ACD Shoulder Height and ACD Shoulder Attenuation, as demonstrated in Fig-

ure 13.1. ACD Shoulder Height is a measure of how nonlinear the transmitter is

driven prior to predistortion whilst ACD Shoulder Attenuation is a measure of dis-

tortion reduction. Since ACD Shoulder Height sets the difficulty of the linearization

problem, the two quantities must be considered as a [Height /Attenuation ] pair. For

example, [ -30 dBc / 15 dB ] is considered better performance than [ -40 dBc / 15 dB ]

since the former is a more difficult linearization problem.

Our performance review spans academic and industry sources including Re-

search Journals, Textbooks, Application Notes, Transmitter Manufacturers, Trans-

mitter Operating Manuals and Transmission Service Providers. From these sources,

we only consider wideband, hardware tested and adaptive predistortion techniques.

This criteria ensures the derived quantitative performance baseline is an accurate

and unbiased representation of predistortion state-of-the-art.

For clarification, we do not consider narrowband, model tested or nonadaptive

techniques for the following reasons. Narrowband techniques demonstrate biased

performance since they do not need to compensate for complex dynamic amplifier

memory [189, 216, 222]. Model tested techniques also demonstrate biased perfor-

mance since their computer models can, at best, only approximate a real wideband

amplifier’s complex dynamic memory [181, 209, 244]. Nonadaptive techniques are

unable to maintain long-term spectral mask compliancy due to transmitter nonlin-

113

CHAPTER 13. PERFORMANCE BASELINE 114

LD

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ACD ShoulderHeight (dBc)

ACD ShoulderAttenuation (dB)

BeforePredistortion

AfterPredistortion

Figure 13.1: Adjacent Channel Distortion (ACD) Shoulder Height and Attenuation

ear drift [50,107,288]. As such they cannot be commercially deployed and hence do

not represent state-of-the-art.

With the findings from each review source presented in the following sections,

it will be seen that different practical circumstances make some sources more pro-

ductive than others. In the end, that current performance identified to be best will

subsequently represent our performance baseline.

13.1 Research Journals

As discussed in Section 2.2 of the literature review, Self-Learning is the most preva-

lent wideband predistortion strategy. Of the many papers written on Self-Learning,

only a small minority present hardware tested performance results. These are sum-

marized in Table 13.1. As can be seen from the third column, we were unable to

identify hardware tested cases for DVB-T or DAB, despite all three target mod-

ulations being equally represented in the literature. It can only be assumed that

cellular hardware is more readily available in academic circles.


Case Paper Signal Shoulder Shoulder Comments

Modulation Height Atten.

(dBc) (dB)

1 [288] 1-carrier WCDMA -43 15 Adaptive �

2 [96] 8 tones over 4MHz -20 15 Low CF Test Signal �

3 [186] 15MHz CDMA -40 16 Nonadaptive �

4 [55] 3-carrier WCDMA -35 20 Nonadaptive �



3-carrier WCDMA -34 16 Nonadaptive �

Table 13.1: Wideband, hardware tested performance as presented in research papers

Of the six wideband, hardware tested cases identified in Table 13.1, only Case 1

can be considered relevant in terms of deriving a quantitative performance baseline

for our technique. Here, the predistorter is adaptive and exhibits [ -43 dBc / 15 dB ]

performance. The remaining cases are ruled out due to inappropriate test signaling

or inability to adapt, as discussed below:

Case 2 - an eight-tone test signal is used to simulate WCDMA. Despite possessing

similar bandwidths, the test signal’s Crest Factor is approximately 3 dB lower

than that of real WCDMA and hence doesn’t truly replicate the high Crest

Factor demands placed on current wideband communication systems. To be

precise, the Crest Factor of this eight-tone test signal is on par with previous

generation 16 QAM linear signal modulation (please refer to Table 1.1’s Crest

Factor comparisons on Page 7). This leads to an over-relaxed predistortion

problem and generously biased test results; considering the very high -20 dB

shoulder height. If this paper had used real WCDMA test signals, we would

have considered its performance in setting our baseline.

Cases 3 & 4 - the algorithms are based on the standard Indirect Learning strategy.

As discussed in Section 2.2.2 of the literature review, this strategy can only

perform adaption by turning off the predistortion filter whilst on-air. This is

specifically to allow fresh transmitter data to flow through to the output and

be collected for postdistortion linear regression. Turning off the predistorter

whilst on-air inevitably leads to unlinearized, high distortion transmission and


hence violation of legal spectral masks and heavy financial fines from regula-

tors. Transmitter operators are not willing to be fined and hence such algo-

rithms are considered commercially unviable. If the algorithms were seamlessly

adaptive, we would have considered their performance in setting our baseline.

Cases 5 & 6 - the algorithms are based on a variation of the standard Indirect

Learning strategy. Whilst this variation sees postdistortion parameters com-

puted via a combination of linear regression and model inverses, instead of

purely the former, the translation process is identical to that of standard

Indirect Learning and hence the strategy suffers the same consequences as

Cases 3 & 4 above. That is, adaption can only be performed by turning

off the predistortion filter whilst on-air, leading to violation of legal spectral

masks and hence commercial unviability. If the algorithms were seamlessly

adaptive, we would have considered their performance in setting our baseline.

Like most readers will be, we are slightly surprised by the lack of wideband and hard-

ware tested and adaptive performance data presented in research journals. It can

only be assumed that the financial cost of transmission hardware and test instrumen-

tation poses a significant constraint for such researchers. Despite this unexpected

finding, the [ -43 dBc / 15 dB ] performance of Case 1 is a good starting point in our

quest to derive a quantitative performance baseline for our technique.

13.2 Textbooks

As a whole, cellular and broadcast textbooks are more oriented towards theo-

retical concepts than applied performance. We did however manage to find one

textbook which was more application related and touched on the subject. This

book, Digital Video and Audio Broadcasting Technology: A Practical Engineering

Guide [72], written by Walter Fischer and sponsored by Rohde & Schwarz, states

that [ -30 dBc / 10 dB ] adaptive performances are achieved in practice for DVB-T.

With the author having many years of experience in the industry, we consider this

figure to be highly reliable.

For consistency, it is worthwhile noting that the [ -43 dBc / 15 dB ] performance

identified in the previous Research Journal section logically trends with this newly

identified [ -30 dBc / 10 dB ] performance. That is, the former less difficult lineariza-

tion problem (lower ACD shoulder height) should, and indeed does, result in greater

ACD shoulder attenuation. This builds confidence in our initial findings.


13.3 Application Notes

Despite many application notes being written on DVB-T andWCDMA, with focused

sections on spectral masks and ACD shoulder measurement, only one was found that

spoke directly about predistortion performance. The Rohde & Schwarz application

note Measurements on MPEG2 and DVB-T Signals - Part 3 [95], written by Sigmar

Grunwald, states that adaptive performances around [ -30 dBc / 10 dB ] are practi-

cally achievable for DVB-T. This is consistent with the previous section’s findings

which also had links to Rohde & Schwarz.

13.4 Transmitter Manufacturers

With success in the previous two sections linked to a transmitter manufacturer, we

felt further success may eventuate from direct contact with other global transmit-

ter manufacturers, specifically NEC, Harris, Ericsson and Nokia Siemens. While

the idea was sound, the result wasn’t. On each occasion, we were advised that

such information was Commercial-In-Confidence and it could not be disclosed. This

response was consistent with their product marketing brochures lacking concrete

performance figures. From this pursuit, it was obvious that Rohde & Schwarz was

more transparent than others when it came to product performance.

13.5 Transmitter Operating Manuals

With no assistance coming from the transmitter manufacturers, we turned atten-

tion to their transmitter operating manuals, hoping that example performance data

would be inter-dispersed with instruction sets. However, after acquiring manuals for

the NEC DTU-31 5KW UHF transmitter [194], NEC DTV-40 2.5KW VHF trans-

mitter [193] and Ericsson 2206 Radio Basestation [61], it soon became apparent that

performance non-disclosure was still being silently enforced. Only basic instructions

on how to set and arm the predistorter from the transmitter’s front panel interface

were presented.

13.6 Transmission Service Providers

Unsatisfied with the previous two outcomes and still determined to acquire applied

performance data, we contacted local transmission service providers, asking if we

could monitor the performance of their in-service systems. Two responded positively,

these being Channel 7 and Broadcast Australia.


Channel 7

Channel 7 granted us access to their Yarrawonga transmitter.1 Consisting of a Ro-

hde & Schwarz SV702 exciter and VH6010A2 amplifier, this transmitter broadcasts

732.5MHz DVB-T at 25W EIRP. Accompanied by Channel 7 radio technician Ian

Smart, the on-air amplifier output spectrum was measured with and without pre-

distortion, via test port coupling. These measurements are presented below.

-120 dBm

-110 dBm

-100 dBm

-90 dBm

-80 dBm

-70 dBm

-60 dBm

-50 dBm

-40 dBm

-130 dBm

-30 dBm

CF 732.5 MHz Span 35.0 MHz

***

RBWVBWSWT

30 kHz300 kHz2 sRef -30.0 dBm

Att 0 dB

Without Predistortion

WithPredistortion

Figure 13.2: Channel 7 Yarrawonga amplifier output

With each trace labeled, we observe a performance of [ -31 dBc / 5 dB ]. Ian Smart

further advised that this was a 1st generation digital transmission system and there-

fore predistortion performance was more than likely not state-of-the-art considering

the processing refinements of later 2nd generation systems. It can only be assumed

that the Rohde & Schwartz linked performance data obtained previously, with 5 dB

better shoulder attenuation, applies to the latest 2nd generation systems.

1Yarrawonga transmission site services the North Ward area of Townsville.


Broadcast Australia

Whilst Broadcast Australia, couldn’t grant us measurement access to any transmit-

ters, they did provide us with pre-measured performance data for Mt Stuart’s ABC

and SBS transmitters.2 The ABC transmitter, considered a 1st generation digital

transmission system, is comprised of a Tandberg exciter and NEC power amplifier

and broadcasts 550.5MHz DVB-T at 200KW EIRP. The SBS transmitter, con-

sidered a 2nd generation digital transmission system, is fully NEC and broadcasts

592.5MHz DVB-T at the same EIRP.

As presented in Figures 13.3 and 13.4, the data provided by Broadcast Australia

consists of the exciter and amplifier output spectra, in both cases with predistor-

tion activated. No data for the predistortion deactivated case was available however.

Raising this with Broadcast Australia, we were advised that since these transmitters

are very high power, on-air deactivation of the predistorter requires regulatory con-

sultation, due to high power spectral mask failure, and is therefore avoided except

for very special circumstances. As in the lower power Yarrawonga case, having this

extra data would have allowed us to define precise performance figures. However all

is not lost.

As discussed in Chapter 7 and specifically demonstrated in Figure 7.6 (page 69),

the predistorter’s ACD Shoulder Height must be greater than the amplifier’s un-

linearized ACD Shoulder Height if self-generated parasitic distortion is to be can-

celed. This means that the exciter ACD Shoulder Heights observed in Figures 13.3a

and 13.4a represent an upper bound on the amplifiers’ unlinearized ACD Shoulder

Heights (the missing data) and therefore bounded performance can be defined as

[< -28 dBc /< 7 dB ] for the ABC transmitter and [< -26 dBc / < 12 dB ] for the SBS

transmitter. With a very conservative 3 dB bounding buffer estimate, derived from

Figure 7.6’s testbed measurements (page 69), these performances are estimated to

be [ -31 dBc / 4 dB ] for ABC and [ -29 dBc / 9 dB ] for SBS.

Before leaving this section, it is worth noting that Lanecomm Communications,

contracted to Kordia Solutions to support the North Queensland Vodafone network,

were also very keen to assist with gathering performance data, specifically WCDMA.

However, despite supervised access being granted to local Vodafone basestations, it

was impossible to take on-air transmission measurements without interfering with

network operation. Quite astonishingly, the Ericsson transmitters in use didn’t

possess external test ports or output coupling. Lanecomm also provided engineering

contacts within Kordia and Telstra but none were able to provide assistance.

2Mt Stuart transmission site services the wider Townsville area.

B

**RBW 30 kHzVBW 300 HzSWT 2.25 sAtt 35 dBRef 8 dBm

Center 550.5 MHz Span 20 MHz2 MHz/

-90

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0

1

Marker 1 [T1 ] -25.15 dBm 550.500000000 MHz

2

Delta 2 [T1 ] -28.06 dB 3.800000000 MHz

3

Delta 3 [T1 ] -27.94 dB -3.800000000 MHz

(a) Exciter output with predistortion

B

**RBW 30 kHzVBW 300 HzSWT 2.25 sRef 17 dBm Att 45 dB


-80

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0

10

1

Marker 1 [T1 ] -15.94 dBm 550.500000000 MHz

2

Delta 2 [T1 ] -35.11 dB 3.800000000 MHz

3

Delta 3 [T1 ] -35.32 dB -3.800000000 MHz

(b) Amplifier output with predistortion

Figure 13.3: ABC Mt Stuart transmission

B

**RBW 30 kHzVBW 300 HzSWT 2.25 sAtt 40 dBRef 12 dBm


-80

-70

-60

-50

-40

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-20

-10

0

10

1

Marker 1 [T1 ] -20.97 dBm 592.500000000 MHz

2

Delta 2 [T1 ] -26.47 dB 3.800000000 MHz

3

Delta 3 [T1 ] -26.18 dB -3.800000000 MHz

(a) Exciter output with predistortion

B

**RBW 30 kHzVBW 300 HzSWT 2.25 sRef 17 dBm Att 45 dB


-80

-70

-60

-50

-40

-30

-20

-10

0

10

1

Marker 1 [T1 ] -16.03 dBm 592.500000000 MHz

2

Delta 2 [T1 ] -37.36 dB 3.800000000 MHz

3

Delta 3 [T1 ] -38.11 dB -3.800000000 MHz

(b) Amplifier output with predistortion

Figure 13.4: SBS Mt Stuart transmission


13.7 Derivation of Performance Baseline

Our review clearly demonstrates the nontriviality associated with obtaining wide-

band and hardware tested and adaptive predistortion performance data. For conve-

nience, its findings are summarized below.

Source Modulation Performance Comment

Research Journal [288] WCDMA [ -43 dBc / 15 dB ] Lowest ACD Shoulder Height

Textbook [72] DVB-T [ -30 dBc / 10 dB ] Linked To Rohde & Schwarz

Application Note [95] DVB-T [ -30 dBc / 10 dB ] Linked To Rohde & Schwarz

Channel 7 DVB-T [ -31 dBc / 5 dB ] 1st Generation Rohde & Schwarz

ABC DVB-T [ -31 dBc / 4 dB ] 1st Generation Tandberg /NEC

SBS DVB-T [ -29 dBc / 9 dB ] 2nd Generation NEC

Table 13.2: Summary of wideband, hardware tested, adaptive performance data

We immediately observe that the majority of reported performances are for DVB-T.

It is stressed that this is a natural outcome of the review with all target modulations

being considered equally in the review process. While more WCDMA and DAB

performances would have been ideal for the sake of completeness, a performance

baseline derived from current best DVB-T performance is more than sound. This

is because DVB-T has the widest bandwidth, highest Crest Factor and highest

amplifier drive level 3 of all three target modulations and is therefore considered

the more difficult linearization problem. If our predistortion technique outperforms

others based on DVB-T, it is guaranteed to outperform others based on WCDMA

and DAB.

For all DVB-T cases, the ACD Shoulder Height is approximately -30 dBc. For

the Channel 7 and ABC 1st generation systems, ACD Shoulder Attenuation is ap-

proximately 5 dB. For the SBS 2nd generation system, ACD Shoulder Attenuation

increases to 9 dB. This 2nd generation NEC performance is consistent with the two

other reports linked to Rohde & Schwarz in the literature.

With two different DVB-T transmitter manufacturers (NEC and Rohde & Schwarz)

reporting the same figures, we are confident that [ -30 dBc / 10 dB ] is the current best

wideband predistortion performance and as such, it is set to be our performance

baseline.

3high power broadcast demands greater amplifier efficiency


For clarification, the lone [ -43 dBc / 15 dB ] WCDMA performance reported in

the first row of Table 13.2 logically trends with this newly derived [ -30 dBc / 10 dB ]

performance baseline. That is, the WCDMA case is a less difficult linearization

problem with its lower ACD shoulder height and therefore it should, and indeed

does, result in greater ACD shoulder attenuation. This logical trending builds further

confidence in our derived baseline.

This baseline, to be used in the next chapter’s performance testing, represents

the minimum level of performance our technique must achieve if it is to be considered

state-of-the-art.

Chapter 14

Performance Of The Proposed

Predistortion Technique

In this chapter, performance of the proposed predistortion technique is tested on

the laboratory transmitter testbed. This testbed was initially discussed in Chap-

ter 4 and hence familiarity is assumed in the following. Performance of the Initial

Setting optimization phase is tested first, with results compared against the previ-

ous chapter’s performance baseline. Performance testing of the On-Air Adaption

phase then follows, with full-cycle disturbance behavior being investigated. In both

cases, internal optimizer parameter settings are in accordance with Section 11.4. Fi-

nally, predistortion Crest Factor growth is discussed, highlighting the need for high

resolution reconstruction DACs.

14.1 Initial Setting Performance Testing

As discussed in Chapter 10, the Initial Setting optimization phase estimates the

optimal predistortion filter parameters ho at the very start of the transmitter’s

operational life. Its guiding optimization schedule, developed in Sections 10.4 and

11.5, is repeated in Table 14.1 for reference. In order to test the Initial Setting

phase, the following steps are performed:

1. Initialize the predistortion filter kernel to h = 0

2. Adjust the power amplifier’s input power to realize a -30 dBc ACD shoulder

height, consistent with the previous chapter’s performance baseline

3. Measure the power amplifier output spectrum prior to predistortion

4. Implement the Initial Setting optimization schedule of Table 14.1

5. Measure the power amplifier output spectrum after Initial Setting

124

CHAPTER 14. PERFORMANCE OF THE PROPOSED TECHNIQUE 125



1b. p3I Genetic


2b. [p5I ∣ p7I ∣ p9I ]


3b. p3I Nelder-Mead

4a. [p5R ∣ p7R ∣ p9R ] SimplexQuadratic, WE = 102

4b. [p5I ∣ p7I ∣ p9I ]

Table 14.1: Initial Setting optimization schedule

Figure 14.1 presents the output power spectra from steps 3 and 5 for each target

modulation. ACP, WACP and Co-Channel Power (CCP) measurements are added to

each subfigure for reference. As discussed in Chapter 3, the carrier frequency for each

target modulation is set in accordance with the ACMA RF spectrum plan [13, 14]

and licensed provider allocation within Australia [7, 15].

Figure 14.2 also presents the output power spectra from steps 3 and 5, but this

time with slots of inband spectral power temporarily removed to uncover Co-Channel

Distortion (CCD). For DVB-T and DAB, two sets of OFDM carriers are nulled dur-

ing modulation whilst for WCDMA, dual notch filtering is applied directly to the

modulated signal. In all cases, slot widths are kept to a minimum in an attempt to

reduce signal degradation and hence maintain experimental comparison with Fig-

ure 14.1. Despite this measure being taken, slight changes in ACD are still witnessed

across figures; specifically slight reduction prior to predistortion due to reduced am-

plifier drive and slight growth after Initial Setting due to optimality offset.

In the previous chapter, predistortion performance was quantified in the frequency-

domain by the [ ACD Shoulder Height, ACD Shoulder Attenuation ] pair. Refer-

ring to Figure 14.1, the proposed technique is seen to achieve a performance of

[ -30 dBc, 13 dB ] for both DVB-T and WCDMA, and [ -30 dBc, 18 dB ] for DAB. At

first glance, these Figure 14.1 results suggest that our technique under performs for

WCDMA, since its [ Height, Attenuation ] performance does not exceed DVB-T’s

and residual CCD shoulders remain after Initial Setting. This is not the case how-

ever for the following reasons:

LD

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prior topredistortion

after InitialSetting

prior afterto Initialpredis. Setting

ACP (dBm) -20.32 -29.47 WACP 172.39 13.61 CCP (dBm) 12.99 13.56

(a) DVB-T

LD

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(b) WCDMA

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(c) DAB

Figure 14.1: Power spectra at the testbed amplifier outputprior to predistortion and after Initial Setting for DVB-T (a),WCDMA (b) and DAB (c).

LD

2VIEW

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(a) DVB-T

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(b) WCDMA

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(c) DAB

Figure 14.2: The case of Figure 14.1 repeated, this time withslots of inband spectral power temporarily removed to un-cover CCD.


1. As initially discussed in Section 4.6, our WCDMA signal comprises 512 user

channels, double the standard 256. This increases the WCDMA signal’s Crest

Factor to that of DVB-T (≈ 11.25 dB), creating a comparable predistortion

problem and resulting in identical [ -30 dBc, 13 dB ] performance. It is only

when the WCDMA signal possesses the standard number of user channels

that performance can be expected to exceed that of DVB-T.

2. Despite our DVB-T and WCDMA test signals possessing comparable Crest

Factors, DVB-T still possesses greater bandwidth and hence experiences greater

Crest Factor growth during predistortion. In order to accommodate this

greater growth, the DVB-T signal must be scaled more heavily prior to entering

the reconstruction DACs. Heavier scaling, coupled with finite DAC precision,

subsequently leads to a higher spectral noise floor; in our case -46 dBc for

DVB-T compared to -52 dBc for WCDMA. The existence of this high noise

floor is also supported by Figures 13.2, 13.3 and 13.4 (previous chapter), rep-

resenting the performance of current in-service systems.

Referring to DVB-T Subfigure 14.1a, this higher noise floor does not allow

predistortion to run to completion; prematurely halting distortion reduction

in the adjacent channel extremities before CCD shoulders can form, as in

WCDMA Subfigure 14.1b. Whilst the noise floor is just low enough for full

CCD reduction, it is not low enough for full ACD reduction and hence a flat

spectral distortion trace appears.

Put another way, if an imaginary -46 dBc noise floor was projected onto WCDMA

Subfigure 14.1b, the residual CCD shoulders would be lost under the noise floor

and performance would appear visually identical to that of DVB-T.1

In effect, if the testbed possessed higher resolution DACs, the noise floor of

Subfigure 14.1a would be lower, allowing complete predistortion, and hence the

after Initial Setting trace would be identical to that of WCDMA; exhibiting

residual CCD shoulders.

3. The existence of residual CCD shoulders in WCDMA Subfigure 14.1b is not a

case of detrimental WACP under-weighting or over-weighting as initially dis-

cussed in Section 9.2. If it were a case of under-weighting, the CCD shoulders

would be concave on the edges of the transmission band, just as the ACD is

prior to predistortion. Alternatively, if it were a case of over-weighting, inner

ACD scalloping would occur and additional shoulders would exist in the outer

adjacent channel regions.

1DVB-T carrier nulling is more efficient than WCDMA notch filtering when it comes to removinginband spectral power and as such, the 1 dB difference in CCD observed between Subfigures 14.2aand 14.2b is a case of notch spectral leakage rather than difference in performance.


These residual CCD shoulders actually represent the practical upper limits of

predistortion linearization, initially spoken of in Section 6.1. Once the amplifier

input drive level reaches CI in the demonstration of Figure 6.1 (page 55), signal

expansion is capped by full output saturation and distortion is inevitable.

So it is seen that when underlying user channels, DAC noise floor considerations and

upper practical linearization limits are taken into account, WCDMA performance is

fully validated.

In the context of the previous Dot-Point 2, it is also suspected that DAB perfor-

mance, with no residual CCD shoulders present, is also limited by the spectral noise

floor. Whilst the noise floor level may be lower here, so too is the difficulty of the

predistortion problem and hence the same scenario is faced. In this case however,

the extent of the limitation is unclear. Unlike the DVB-T case, we don’t have a

comparable WCDMA cross check available and hence it is quite possible that per-

formance greater than [ -30 dBc, 18 dB ] is achievable if higher precision DACs are

used.

In order to put the performance of this technique into context, we turn to the

previous chapter’s performance baseline of [ -30 dBc, 10 dB ]. Derived for DVB-T, the

most difficult target modulation to predistort, this baseline represents the minimum

level of performance our technique must achieve if it is to be considered state-of-

the-art. As demonstrated in Subfigure 14.1a, our technique achieves a performance

of [ -30 dBc, 13 dB ] for DVB-T. This is a 3 dB improvement over the baseline and as

such, the proposed technique must be taken seriously.

It is important to reiterate that the spectral plots of Figures 14.1 and 14.2 are

taken at the output of the power amplifier, and thus bandpass mask filtering is still

to be applied. It is this additional selectivity which brings ACD levels to within

spectral mask requirements. As an example, for DVB-T transmitter collocation,

the Australian spectral mask is set at -48 dBc in the adjacent channel [41]. From

Subfigure 14.1a, ACD after predistortion is between -46 dBc and -43 dBc, ignoring

the limits of the spectral noise floor discussed earlier. As such, at the output of the

amplifier, an additional 2–5 dB attenuation is required in order to satisfy the mask.

With [261] reporting > 40 dB selectivity for its DVB-T bandpass filters, the mask is

well satisfied prior to transmission.

Referring to the added ACP, WACP and CCP measurements of Figure 14.1, it

is seen that reduction in ACP /WACP is accompanied by growth in CCP. This is

not coincidental. As discussed intuitively in Section 6.1, the predistorter’s reversal

of signal compression leads to the restoration of top end transmitter gain and hence

transmitted power.

Another observation of Figure 14.1 worthy of mention is the slight difference in

lower and upper ACD shoulder heights prior to predistortion. As discussed through-


out previous chapters, this spectral asymmetry is an artifact of transmitter memory.

For DVB-T, possessing greatest bandwidth, this asymmetry extends into the outer

adjacent channel, with scalloping witnessed at 7MHz above the carrier frequency.

With the presentation of linearization performance now complete, attention is

turned to convergence rate performance. Table 14.2 presents the convergence rate

properties of the Initial Setting optimization schedule for DVB-T /WCDMA/DAB.

As discussed in Sections A.4 and A.5, the local Nelder-Mead Simplex and global

Genetic algorithms are both robust, gradient-free strategies and hence do not require

objective measurement averaging. For both algorithms and all target modulations,

the spectrum analyzer sweep rate is set to two seconds.

Expectedly, the global Genetic algorithm is much slower to converge than the

local Nelder-Mead Simplex algorithm, with convergence rates being measured in

hours compared to minutes (last column). This makes the first half of the schedule

(search phase) much slower than the last half (refinement phase). Convergence is also

seen to be slowest for DVB-T and quickest for DAB. This is a direct consequence of

the relative number of variables to be optimized (column 4). Total time to converge

is obtained by adding the convergence rates of each optimization in the last column.

For DVB-T /WCDMA/DAB, this turns out to be 25.1 / 21.8 / 9.3 hours.

It is immediately obvious from these convergence rates that the Initial Setting

optimization phase must be left to run over night and / or in the background during

the transmitter commissioning phase. This will not pose a problem however, pro-

vided technicians proactively schedule other on-site work packages to accommodate.

It is acknowledged that existing techniques have quicker convergence rates. For

example the Direct Learning technique converges in the order of several hours [194].

However this quicker convergence rate is at the expense of linearization performance.

Specifically, predistortion filter parameter estimation is a trade off between the two

conflicting criteria of convergence rate and linearization performance.

Compared to existing techniques, the proposed technique occupies the opposite

end of this trade off spectrum. That is, it accepts a slower convergence rate specif-

ically to maximize linearization performance. Such performance has been demon-

strated earlier in this section.

As discussed in the literature review, this trade off stems from the different pa-

rameter estimation models being employed and the validity of their assumptions

relating to objective convexity. Existing techniques use a local linear regression

model whereas the proposed technique uses a global generic single objective mathe-

matical optimization model. The former cannot guarantee true global convergence

whereas the latter can.

Optimization

hs

Optimizer

Variables

Itera

tions

Objective

Converg

ence

Measu

rements

Rate

1a.

p3R

8/7/3

105/97

/37

5300

/4900

/1900

2.9/2.7/1hrs

1b.

p3I

Gen

etic

8/7/3

AsAbove

AsAbove

AsAbove

2a.

[ p5R

∣ p7R

∣ p9R]

24/21

/9

325/276/124

16300/13850/6250

9/7.7/3.5hrs

2b.

[ p5I∣ p

7I∣ p

9I]

24/21

/9

AsAbove

AsAbove

AsAbove

3a.

p3R

8/7/3

86/71

/34

227/177/75

7.5/6/2.5mins

3b.

p3I

Nelder-M

ead

8/7/3

AsAbove

AsAbove

AsAbove

4a.

[ p5R

∣ p7R

∣ p9R]

Sim

plex

24/21

/9

250/215/91

951/762/244

32/25

/8mins

4b.

[ p5I∣ p

7I∣ p

9I]

24/21

/9

AsAbove

AsAbove

AsAbove

⇓

Tota

lTim

eToConverg

e:

25.1

/21.8

/9.3

hrs

Tab

le14.2:Con

vergen

cerate

properties

oftheInitialSettingop

timizationsched

ule

forDVB-T

/WCDMA/DAB


14.2 On-Air Adaption Performance Testing

As discussed in Chapter 10, a transmitter’s nonlinear transfer characteristic will

drift slowly during its operational life. This is a result of component aging, temper-

ature fluctuations and power supply voltage variations. The On-Air Adaption phase

adapts the predistortion filter kernel during this drift, in order to maintain optimal

linearization.

In practice, the On-Air Adaption phase is applied directly after the Initial Setting

phase and left to run for the lifetime of the transmitter. In this sense, kernel adaption

is a continuous ongoing process which constantly tracks changes in the transfer

characteristic. Its guiding optimization schedule, developed in Sections 10.5 and

11.5, is repeated in Table 14.3 for reference.

Since component aging is a long-term process, typically occurring over a duration

of years, it has minimal effect on the short-term operation of the On-Air Adaption

phase. From the perspective of a continuously running optimizer, which is the On-

Air Adaption phase, it is the short-term temperature fluctuations and power supply

voltage variations, typically occurring over a duration of hours or days, which are

the dominant causes of drift and which necessitate kernel adaption.

In the laboratory testing of the On-Air Adaption phase, these temperature and

supply voltage fluctuations are induced to create a forced (as opposed to natural)

drift. Temperature is reduced from a nominal 45℃ to 35℃ (22% change) whilst

supply voltage is reduced from a nominal 28V to 25V (10.7% change) [205]. Due

to the controlled environmental and electrical operating conditions of modern day

transmitters [61, 194], such large changes would never be encountered in practice.

They are chosen here however to test the On-Air Adaption phase under the most

extreme drift conditions. Temperature and supply voltage are intentionally reduced

in this testing in order to reduce FET conductance and compress FET dynamic

range respectively, thereby severely degrading the overall nonlinearity.


1a. p3R

1b. p3I Nelder-Mead Quadratic, WE = 102

2a. [p5R ∣ p7R ∣ p9R ] Simplex

2b. [p5I ∣ p7I ∣ p9I ]


Table 14.3: On-Air Adaption optimization schedule


In the first stage of testing (following completion of the Initial Setting phase),

a forced drift is applied and the On-Air Adaption phase is allowed to react. In the

second stage of testing (continuing directly on from the first), nominal conditions

are restored and the On-Air Adaption phase is once again allowed to react. With

this two stage testing methodology, the performance of the On-Air Adaption phase

can be observed in terms of a full-cycle disturbance rather than a one-way drift.

Figures 14.3 and 14.4 present results for these first and second stages of testing

respectively. Both figures are presented side-by-side on the same page to facilitate

easy comparison.

In Figure 14.3 (first stage testing), output power spectra prior to predistortion,

after Initial Setting, after forced drift and after On-Air Adaption are presented for

each target modulation. ACP, WACP and CCP measurements are added to each

subfigure for reference. The forced drift renders the Initial Setting predistorter ker-

nel non-optimal, resulting in a significant increase in distortion from Initial Setting

levels. To restore kernel optimality under the forced drift conditions, the On-Air

Adaption phase is applied. At first glance, it appears that the On-Air Adaption

phase under performs since its trace does not align with that of the Initial Set-

ting phase. This is not the case however. From an optimization perspective, the

forced drift causes the global minimum of the WACP optimization objective to not

only translate in the vector space but also increase in magnitude. This increase in

magnitude is a direct consequence of the amplifier’s greater inherent nonlinearity

as discussed previously in terms of FET conductance and dynamic range. It fol-

lows that the On-Air Adaption phase, under these extreme drift conditions, cannot

theoretically reduce distortion back to an Initial Setting level. The best it can do

is reduce distortion to a level corresponding to the new inferior global minimum

and this is what we are observing with the two traces not matching. Difference in

magnitude between the WACP global minimum before and after the forced drift is

frequency and bandwidth dependent and hence On-Air Adaption results will vary

across target modulation.

In Figure 14.4 (second stage testing), output power spectra after first stage On-

Air Adaption (carried over from Figure 14.3), after restoration of nominal conditions

and after second stage On-Air Adaption are presented for each target modulation.

Once again, power measurements are added to each subfigure for reference. Restora-

tion of nominal conditions renders the first stage On-Air Adaption predistorter kernel

non-optimal, resulting in a significant increase in distortion levels. To restore kernel

optimality under the nominal conditions, the second stage On-Air Adaption phase

is applied. This brings distortion levels back to the original Initial Setting levels.

LD

RBW 30 kHz

VBW 300 kHz

SWT 2 s

RF Att 40 dB

Mixer -40 dBm

A

2RM

3RM

4RM

Unit dBm

1VIEW

2VIEW

3VIEW

4VIEW

1RM

Ref Lvl

14.5 dBm

Ref Lvl

14.5 dBm


9.5 dB Offset *

-80

-70

-60

-50

-40

-30

-20

-10

0

10

-85.5

14.5

1) prior topredistortion

3) afterforced drift

4) after 1st stageOn-Air Adaption

prior after after afterto Initial forced 1st stagepredis. Setting drift OA Adapt.

ACP (dBm) -20.32 -29.47 -25.60 -27.86 WACP 172.39 13.61 40.34 20.01 CCP (dBm) 12.99 13.56 13.75 13.85

2) after InitialSetting

(a) DVB-T

LD

RBW 30 kHz

VBW 300 kHz

SWT 2 s

RF Att 40 dB

Mixer -40 dBm

A

2RM

3RM

4RM

Unit dBm

1VIEW

2VIEW

3VIEW

4VIEW

1RM

Ref Lvl

12 dBm

Ref Lvl

12 dBm


7 dB Offset *

-80

-70

-60

-50

-40

-30

-20

-10

0

-88

12







(b) WCDMA

LD

Ref Lvl

16 dBm

Ref Lvl

16 dBm

RBW 30 kHz

VBW 300 kHz

SWT 2 s

RF Att 50 dB


1 dB Offset

Mixer -40 dBm

A

2RM

3RM

4RM

Unit dBm

1VIEW

2VIEW

3VIEW

4VIEW

1RM

*

-80

-70

-60

-50

-40

-30

-20

-10

0

10

-84

16







(c) DAB

Figure 14.3: First stage of On-Air Adaption testing forDVB-T (a), WCDMA (b) and DAB (c). Output power spec-tral traces individually labeled.

LD

Ref Lvl

14.5 dBm

Ref Lvl

14.5 dBm

RBW 30 kHz

VBW 300 kHz

SWT 2 s

RF Att 40 dB


9.5 dB Offset

Mixer -40 dBm

A

1RM

3RM

4RM

Unit dBm

1VIEW

2VIEW

3VIEW

4VIEW

2RM

*

-80

-70

-60

-50

-40

-30

-20

-10

0

10

-85.5

14.5

2) after nominalrestoration

3) after 2nd stageOn-Air Adaption


after after after after1st stage nominal 2nd stage InitialOA Adapt. rest. OA Adapt. Setting

ACP (dBm) as prev. -26.65 -29.41 as prev. WACP as prev. 31.12 13.69 as prev. CCP (dBm) as prev. 13.38 13.51 as prev.


(a) DVB-T

LD

1VIEW

2VIEW

3VIEW

4VIEW

Ref Lvl

12 dBm

Ref Lvl

12 dBm

RBW 30 kHz

VBW 300 kHz

SWT 2 s

RF Att 40 dB


7 dB Offset

Mixer -40 dBm

A

1RM

2RM

3RM

4RM

Unit dBm

*

-80

-70

-60

-50

-40

-30

-20

-10

0

-88

12







(b) WCDMA

LD

Ref Lvl

16 dBm

Ref Lvl

16 dBm

RBW 30 kHz

VBW 300 kHz

SWT 2 s

RF Att 50 dB


1 dB Offset

Mixer -40 dBm

A

1RM

2RM

3RM

Unit dBm

1VIEW

2VIEW

3VIEW

4VIEW 4RM

*

-80

-70

-60

-50

-40

-30

-20

-10

0

10

-84

16







(c) DAB

Figure 14.4: Second stage of On-Air Adaption testing forDVB-T (a), WCDMA (b) and DAB (c). Output power spec-tral traces individually labeled.


The fact that distortion levels rise following a change in operating condition (first

stage forced drift or second stage nominal restoration) but are subsequently reduced

by application of the On-Air Adaption phase demonstrates the optimal tracking

ability of the On-Air Adaption phase. The fact that the On-Air Adaption phase is

capable of returning the predistorter kernel to its Initial Setting state following the

full-cycle disturbance also demonstrates reliable and predictable behavior.

With the On-Air Adaption phase performed by the local Nelder-Mead Simplex

algorithm, convergence rates for both first and second stage testing above are in the

order of minutes for all target modulations. This rate is more than adequate, con-

sidering transmitter drift is continuous in practice with a much larger hourly / daily

time frame.

Before leaving this section, Tables 14.4–14.6 present predistortion filter ker-

nel coefficients for DVB-T, WCDMA and DAB signal modulation, following the

Initial Setting and first stage On-Air Adaption phases. Three trends are immedi-

ately obvious from these tables:

1. Real components are generally an order of magnitude larger than imaginary

components. This suggests that imaginary components perform a less dom-

inant role in the dynamic memory compensation process. As discussed in

Chapter 10, this offset magnitude is the reason why real components are op-

timized before imaginary components in both the Initial Setting and On-Air

Adaption phases.

2. Coefficient magnitude reduces by several orders of magnitude per odd non-

linear order. As discussed in Chapters 10 and 11, this poor scaling instigates

the development of the optimization scheduling concept and forces the Nelder-

Mead Simplex and Genetic algorithms to employ coefficient pre-scaling during

simplex initialization and gene mutation respectively.

3. Coefficient magnitude reduces globally when moving from DVB-T to WCDMA

to DAB. This is due to a decrease in modulation bandwidth and hence total

predistortion power needing to be generated for linearization.

It is also interesting to observe that, despite amplifier memory effects decaying with

time, coefficient magnitudes generally hold steady with memory delay i. Considering

the direct memory estimation technique of Section 12.2 and the positive predistortion

performances already presented in this chapter, this phenomenon is not attributed

to memory length under-estimation. Rather, what we are observing is the optimizer

actively compensating for the non-dominant coefficients lost during pruning; specif-

ically increasing and potentially phase offsetting those back end kernel coefficients

which would normally decay with memory.

DVB-T KERNEL COEFFICIENTS

Kernel Memory Coefficient After Coefficient After Change In

Order Delay i Initial Setting 1st Stage On-Air Adaption Coefficient

M = 14,R = 2 Real Imag Real Imag Real Imag

0 7.311 × 10−7 1.645 × 10−8 1.041 × 10−6 2.409 × 10−7 −3.099 × 10−7 −2.240 × 10−7

1 −1.019 × 10−7 1.739 × 10−8 −3.000 × 10−7 3.766 × 10−8 1.981 × 10−7 −2.027 × 10−8

2 −3.088 × 10−7 −4.111 × 10−7 −4.372 × 10−7 −5.569 × 10−7 1.284 × 10−7 1.458 × 10−7

3 3 −5.538 × 10−7 1.944 × 10−8 5.812 × 10−7 2.713 × 10−8 −5.538 × 10−7 −7.690 × 10−9

4 5.716 × 10−7 3.613 × 10−8 7.607 × 10−7 1.069 × 10−7 −1.891 × 10−7 −7.077 × 10−8

5 −2.509 × 10−7 2.530 × 10−8 −4.295 × 10−7 −3.926 × 10−8 1.786 × 10−7 6.456 × 10−8

6 −2.901 × 10−7 −9.742 × 10−8 −3.554 × 10−7 −1.023 × 10−8 6.530 × 10−8 −8.719 × 10−8

7 2.456 × 10−8 3.697 × 10−8 3.270 × 10−7 7.074 × 10−8 −3.024 × 10−7 −3.377 × 10−8

0 −5.481 × 10−13 −1.000 × 10−13 1.513 × 10−13 −5.865 × 10−14 −6.994 × 10−13 −4.135 × 10−14

1 −3.861 × 10−13 −2.402 × 10−14 −3.741 × 10−13 −6.835 × 10−14 −1.200 × 10−14 4.433 × 10−14

2 1.150 × 10−13 −2.515 × 10−14 9.218 × 10−14 −4.759 × 10−14 2.282 × 10−14 2.244 × 10−14

5 3 −2.941 × 10−13 2.070 × 10−15 3.102 × 10−13 4.271 × 10−15 −6.043 × 10−13 −2.201 × 10−15

4 −8.944 × 10−14 −5.892 × 10−13 −3.091 × 10−13 −2.351 × 10−14 2.197 × 10−13 −5.657 × 10−13

5 2.964 × 10−13 −7.423 × 10−14 −5.351 × 10−14 4.122 × 10−14 3.499 × 10−13 −1.155 × 10−13

6 1.943 × 10−13 5.576 × 10−14 8.593 × 10−14 −9.292 × 10−15 1.084 × 10−13 6.505 × 10−14

7 −3.518 × 10−14 −4.955 × 10−14 7.336 × 10−13 −3.258 × 10−14 −7.688 × 10−13 −1.697 × 10−14

0 4.528 × 10−19 3.986 × 10−20 5.667 × 10−19 8.842 × 10−20 −1.139 × 10−19 −4.856 × 10−20

1 −1.181 × 10−19 2.599 × 10−19 −2.630 × 10−19 −3.558 × 10−20 1.449 × 10−19 2.955 × 10−19

2 3.392 × 10−19 −3.370 × 10−20 3.587 × 10−19 −5.138 × 10−20 −1.950 × 10−20 1.768 × 10−20

7 3 1.108 × 10−19 2.973 × 10−19 1.583 × 10−19 2.896 × 10−19 −4.750 × 10−20 7.700 × 10−21

4 −4.831 × 10−19 5.032 × 10−20 −4.965 × 10−19 9.633 × 10−20 1.340 × 10−20 −4.601 × 10−20

5 −8.751 × 10−19 2.458 × 10−20 −3.276 × 10−19 9.935 × 10−20 −5.475 × 10−19 −7.477 × 10−20

6 4.435 × 10−20 −1.032 × 10−19 6.833 × 10−19 2.425 × 10−20 −6.390 × 10−19 −1.275 × 10−19

7 3.366 × 10−19 7.425 × 10−20 −9.523 × 10−20 −4.476 × 10−20 4.318 × 10−19 1.191 × 10−19

0 4.617 × 10−24 2.270 × 10−24 4.947 × 10−24 2.260 × 10−24 −3.300 × 10−25 1.000 × 10−26

1 −5.556 × 10−24 −2.854 × 10−25 −5.378 × 10−24 −2.983 × 10−25 −1.780 × 10−25 1.290 × 10−26

2 6.532 × 10−24 −3.229 × 10−25 3.232 × 10−24 −3.525 × 10−25 3.300 × 10−24 2.960 × 10−26

9 3 2.028 × 10−24 1.010 × 10−25 2.086 × 10−24 1.304 × 10−25 −5.800 × 10−26 −2.940 × 10−26

4 −7.416 × 10−25 −1.816 × 10−25 9.858 × 10−25 −3.805 × 10−24 −1.727 × 10−24 3.623 × 10−24

5 −5.725 × 10−24 3.665 × 10−24 −5.438 × 10−24 −2.488 × 10−25 −2.870 × 10−25 3.914 × 10−24

6 3.887 × 10−24 8.256 × 10−25 3.375 × 10−24 −2.574 × 10−25 5.120 × 10−25 1.083 × 10−24

7 9.736 × 10−25 −3.943 × 10−25 1.454 × 10−24 7.466 × 10−25 −4.804 × 10−25 −1.141 × 10−24

Table 14.4: Predistortion filter kernel coefficients following Initial Setting and firststage On-Air Adaption for DVB-T.

WCDMA KERNEL COEFFICIENTS




0 7.845 × 10−5 1.675 × 10−6 1.021 × 10−4 2.133 × 10−5 −2.365 × 10−5 −1.966 × 10−5

1 1.291 × 10−5 −1.020 × 10−5 −7.173 × 10−5 −6.892 × 10−6 8.464 × 10−5 −3.308 × 10−6

2 −2.963 × 10−5 −5.636 × 10−6 −4.878 × 10−5 −1.135 × 10−5 1.915 × 10−5 5.714 × 10−6

3 3 2.406 × 10−5 −2.320 × 10−6 4.808 × 10−5 5.301 × 10−6 −2.402 × 10−5 −7.621 × 10−6

4 −5.390 × 10− 7 9.610 × 10−6 2.349 × 10−5 6.339 × 10−7 −2.403 × 10−5 8.976 × 10−6

5 7.266 × 10−5 −5.997 × 10−6 −4.694 × 10−6 −2.424 × 10−6 7.735 × 10−5 −3.573 × 10−6

6 3.475 × 10−6 7.462 × 10−7 5.358 × 10−5 −3.337 × 10−6 −5.011 × 10−5 4.083 × 10−6

0 −8.780 × 10−9 −2.809 × 10−10 −3.576 × 10−9 −9.327 × 10−11 −5.204 × 10−9 −1.876 × 10−10

1 9.950 × 10−9 −5.558 × 10−10 4.724 × 10−9 −3.148 × 10−10 5.226 × 10−9 −2.410 × 10−10

2 1.166 × 10−9 −1.230 × 10−9 −4.836 × 10−10 −3.696 × 10−9 1.650 × 10−9 2.466 × 10−9

5 3 −6.614 × 10−9 2.639 × 10−10 −7.551 × 10−9 7.799 × 10−11 9.370 × 10−10 1.859 × 10−10

4 −1.629 × 10−9 5.182 × 10−11 2.217 × 10−9 −1.878 × 10−10 −3.846 × 10−9 2.396 × 10−10

5 −7.978 × 10−10 −1.347 × 10−10 −4.321 × 10−10 −4.181 × 10−10 −3.657 × 10−10 2.834 × 10−10

6 3.546 × 10−10 −4.335 × 10−11 7.336 × 10−10 −2.743 × 10−10 −3.790 × 10−10 2.310 × 10−10

0 −1.780 × 10−13 9.245 × 10−14 2.064 × 10−12 5.794 × 10−13 −2.242 × 10−12 −4.870 × 10−13

1 1.581 × 10−12 2.532 × 10−13 5.063 × 10−12 5.434 × 10−13 −3.482 × 10−12 −2.902 × 10−13

2 8.150 × 10−13 4.743 × 10−13 2.957 × 10−12 1.074 × 10−12 −2.142 × 10−12 −5.997 × 10−13

7 3 −2.071 × 10−12 1.520 × 10−13 1.300 × 10−12 9.635 × 10−13 −3.371 × 10−12 −8.115 × 10−13

4 2.315 × 10−12 5.207 × 10−13 3.152 × 10−12 1.515 × 10−13 −8.370 × 10−13 3.692 × 10−13

5 6.182 × 10−13 6.323 × 10−13 1.399 × 10−12 1.934 × 10−11 −7.808 × 10−13 −1.871 × 10−11

6 4.476 × 10−13 2.894 × 10−14 3.836 × 10−12 8.449 × 10−13 −3.388 × 10−12 −8.160 × 10−13

0 9.482 × 10−16 1.482 × 10−17 3.471 × 10−16 5.285 × 10−16 6.011 × 10−16 −5.137 × 10−16

1 −1.602 × 10−16 −1.124 × 10−16 5.170 × 10−16 4.201 × 10−17 −6.772 × 10−16 −1.544 × 10−16

2 −7.352 × 10−16 −3.624 × 10−17 −5.935 × 10−16 −3.827 × 10−17 −1.417 × 10−16 2.030 × 10−18

9 3 −2.483 × 10−16 −6.708 × 10−17 −1.066 × 10−15 −2.575 × 10−16 8.177 × 10−16 1.904 × 10−16

4 −7.842 × 10−17 −8.686 × 10−17 −9.434 × 10−16 −2.629 × 10−16 8.650 × 10−16 1.760 × 10−16

5 7.294 × 10−16 −2.039 × 10−16 −2.107 × 10−16 1.543 × 10−17 9.401 × 10−16 −2.193 × 10−16

6 3.774 × 10−16 1.527 × 10−17 9.251 × 10−16 −2.644 × 10−16 −5.477 × 10−16 −2.491 × 10−16

Table 14.5: Predistortion filter kernel coefficients following Initial Setting and firststage On-Air Adaption for WCDMA.

DAB KERNEL COEFFICIENTS




0 1.780 × 10−5 −2.703 × 10−6 1.230 × 10−5 1.490 × 10−5 2.050 × 10−5 −1.760 × 10−5

3 1 −6.580 × 10−5 1.051 × 10−6 3.581 × 10−6 −6.448 × 10−7 −6.938 × 10−5 1.696 × 10−6

2 −6.569 × 10−6 −6.534 × 10−7 −7.822 × 10−5 1.362 × 10−6 7.165 × 10−5 −2.015 × 10−6

0 5.185 × 10−9 3.232 × 10−10 3.739 × 10−9 5.284 × 10−10 1.446 × 10−9 −2.052 × 10−10

5 1 6.506 × 10−9 1.257 × 10−10 5.207 × 10−9 5.309 × 10−10 1.299 × 10−9 −4.052 × 10−10

2 3.142 × 10−9 −4.799 × 10−11 1.810 × 10−9 −1.489 × 10−10 1.332 × 10−9 1.009 × 10−10

0 −1.537 × 10−12 8.552 × 10−13 −4.093 × 10−12 1.129 × 10−12 2.556 × 10−12 −2.738 × 10−13

7 1 1.586 × 10−13 −3.739 × 10−13 2.157 × 10−12 −8.125 × 10−13 −1.999 × 10−12 4.386 × 10−13

2 −9.163 × 10−12 −8.692 × 10−13 1.146 × 10−12 −9.573 × 10−13 −1.031 × 10−11 8.810 × 10−14

0 2.230 × 10−16 8.809 × 10−17 9.939 × 10−16 9.422 × 10−17 −7.709 × 10−16 −6.130 × 10−18

9 1 9.032 × 10−17 −7.966 × 10−17 2.524 × 10−16 4.586 × 10−17 −1.621 × 10−16 −1.255 × 10−16

2 1.577 × 10−16 9.282 × 10−17 2.496 × 10−17 1.426 × 10−16 1.327 × 10−16 −4.978 × 10−17

Table 14.6: Predistortion filter kernel coefficients following Initial Setting and firststage On-Air Adaption for DAB.


14.3 Crest Factor Growth

As discussed in Section 6.1, a digital predistortion system fundamentally implements

a signal expansion characteristic in order to compensate for the power amplifier’s

impending compression. This expansion characteristic logically leads to signal Crest

Factor growth. Taking into account memory, this growth increases with bandwidth.

Figure 14.5 demonstrates this growth for DVB-T, in terms of the Complementary

Cumulative Distribution Function (CCDF). A signal’s CCDF represents the prob-

ability (y-axis) of the signal’s instantaneous power exceeding its mean power by a

specified value (x-axis). As such, the CCDF x-intercept represents Crest Factor.

1E-05

1E-04

1E-03

0.01

0.1

*Att 30 dB

AQT 303.80msRef 20.00 dBm

CF 670.0 MHz Mean Pwr + 20.0 dB

2Sa

View

3Sa

View

4Sa

View

1000000 samples


after 1st stageOn-Air Adaption


Pro

babi

lity

CF 670.0 MHz

0 2010 12 14 16 182 4 6 8

Instantaneous Power Exceeding Mean Power (dB)

Figure 14.5: CCDF of the predistortion filter output signal prior to predistortion,after Initial Setting and after first stage On-Air Adaption for DVB-T modulation.Derived using the envelope power approach and 1 000 000 signal samples [18].

It is seen here that the predistortion filter output signal prior to predistortion is a

pure DVB-T signal with 11.25 dB Crest Factor. After Initial Setting and first stage

On-Air Adaption however, the predistortion filter expands signal peaks and increases

Crest Factor by 3.25 dB and 3.75 dB respectively, to counteract the downstream


compression of the power amplifier. It is noted that signal expansion is greater

in the first stage On-Air Adaption case since power amplifier compression is more

severe following the forced drift.

Crest Factor growth requires headroom preallocation along the exciter stage path

and hence places greater expectations on the dynamic range of exciter components.

Finite precision reconstruction DACs are generally the weakest link in this respect,

as is the case in the laboratory transmitter testbed.

In terms of headroom preallocation, testbed DAC inputs are scaled such that

peak excursion is approximately 30%, 36% and 39% of DAC full scale for DVB-T,

WCDMA and DAB respectively. This staggered scaling accomodates the different

Crest Factor growths associated with bandwidth. After Initial Setting and first

stage On-Air Adaption, peak excursion for all target modulations grows to approx-

imately 75% and 80% of DAC full scale respectively, and hence destructive clipping

is avoided.

In terms of dynamic range, testbed DACs possess a reasonable 14-bit resolution.

With the above scaling and Crest Factor growth however, only a small portion of

these bits are utilized on the average and hence spectral noise floors are driven up

to -46 dBc, -52 dBc and -51 dBc for DVB-T, WCDMA and DAB respectively. As

discussed earlier in Section 14.1, these noise floors are found to interfere with DVB-T

and DAB bottom end predistortion performance and hence testbed DACs turn out

to be several bits short of ideal.

It follows that special consideration must be given to DAC resolution in predis-

tortion systems. What may be adequate for normal transmission will unlikely be

adequate for predistortion transmission, considering the high Crest Factors involved.

In this sense, it is always recommended to use the highest precision DACs available.

This chapter has demonstrated the performance of the proposed predistortion tech-

nique on real hardware, specifically the laboratory transmitter testbed. Initial Setting

results show a 3 dB improvement in performance over what is considered current

state-of-the-art. Full-cycle On-Air Adaption results also show optimal tracking abil-

ity and predictable behavior in beyond worst case disturbance conditions. The need

for high resolution reconstruction DACs is also reinforced in the context of dynamic

range and hence performance potential.

Chapter 15

Summary & Conclusion

Digital predistortion is a modern linearization technique allowing transmitters to be

operated more efficiently and hence cost effectively. In accordance with our State-

ment of Research in Section 3.1, this thesis has specifically demonstrated adaptive

digital predistortion for wideband high Crest Factor applications based on the con-

cept of Spectral Power Feedback Learning (SPFL). Prior to this work, SPFL had

only been proposed for narrowband, linearly modulated systems.

Unlike the current generation Direct and Indirect Learning strategies, which

rely on time-domain signal feedback, the SPFL strategy operates with frequency-

domain information feedback. This makes the strategy more suited to current and

future wideband applications since temporal feedback delay and gain compensa-

tion is avoided. This is consistent with demonstrated state-of-the-art performance

results obtained from real hardware testing of DVB-T, WCDMA and DAB signal

modulations.

Research Contributions

With the predistortion filter’s characterizing parameters interpreted as a set of vari-

ables to be optimized and a measure of output spectral distortion interpreted as a

linearizing optimization objective, the SPFL strategy employs generic mathematical

optimization to estimate predistortion filter parameters. In the context of this op-

timization framework, this research’s wideband evolution of the SPFL strategy has

produced the following contributions to the digital predistortion community:

1. the definition of a new spectral distortion optimization objective, specifically

theWeighted Adjacent Channel Power (WACP). In addition to conveying com-

plete adjacent channel behavior, this objective is able to discriminate between

spectral distortion components and hence control the location of spectral dis-

tortion reduction. This makes linearization more robust in the presence of

residual memory effects.

140

CHAPTER 15. SUMMARY & CONCLUSION 141

2. the definition of a new predistortion filter architecture to accommodate am-

plifier memory effects. Based on hybrid triple-stage pruning of the classic

Baseband Volterra Series, this architecture possesses a dynamically matched

kernel whose size is not only linear with respect to memory, but also inde-

pendent of hardware sampling rate implementation. Ultimately, this kernel

size ensures a practically manageable optimization vector space and hence

improved convergence reliability.

3. the demonstration of a new experimental procedure for estimating predistor-

tion filter memory. This procedure is based on sweep-probing the transmitter

with memory specific distortion created by the predistortion filter, and look-

ing for changes in the signature of the output ACP spectrum. Compared to

traditional approaches, this procedure is considered more direct and accurate

since it estimates memory directly with the predistortion filter in place and

hence replicates what the predistortion filter and transmitter would experience

in practice during optimization.

4. the derivation of Initial Setting and On-Air Adaption optimization schedules

based on the concept of influential subsets. This concept ensures optimization

is always performed over the minimally sized variable vector which guarantees

complete observability and hence optimization convergence reliability is always

maximal.

5. the development of the Distortion Array ; a graphical organizing tool for keep-

ing track of nonlinear distortion components generated by the predistorter-

amplifier cascade. Using this tool, all facets of the predistortion process can

be described both visually and intuitively in the time-domain; this includes

the fundamental concepts of parasitic growth, nonlinear order interaction and

theoretical limitation. Without the Distortion Array, predistortion concepts

are easily lost in mathematical rigor.

The legitimacy of these research contributions is confirmed with two full-length, peer

reviewed journal articles being published in the IEEE Transactions on Broadcasting.

The Way Forward

Considering its demonstrated state-of-the-art performance, we believe the way for-

ward for this technique is technology commercialization. With the assistance of

Uniquest (James Cook University’s commercialization business), we have filed an

international patent application and passed the international search report phase;

highlighting our claims of novelty. We are now seeking commercialization partners

and subsequent funding to progress the patent from provisional to full status.

CHAPTER 15. SUMMARY & CONCLUSION 142

The commercialization partners we intend targeting are digital broadcast and

mobile basestation transmitter manufacturers; specifically Rohde & Schwarz, NEC,

Harris, Ericsson and Nokia Siemens. In approaching these global companies, our

discussions will be focused on the following key points:

• future growth in modulation bandwidth will further expose the feedback weak-

nesses of current generation Direct and Indirect Learning strategies, leading

to reduced linearization performance. The proposed technique on the other

hand, with its novel predistortion filter architecture and parameter estimation

strategy, is far more robust and well suited to future wideband applications.

• the proposed technique’s performance is demonstrated on a working hardware

testbed, eliminating any uncertainty in the validity of results. Furthermore,

this demonstrated performance is shown to be state-of-the-art when baselined

against current in-service wideband predistortion systems.

Based on the solid research leading to this point, we approach the upcoming com-

mercialization effort with vigor and optimism. We truly believe this technique has

the potential to advance digital predistortion application and represent the next

generation of predistortion system.

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Appendix A

Shortlisted Optimization

Algorithms

A.1 Gradient Descent

The Gradient Descent (GD) optimization algorithm is classified as a line search

algorithm [179]. At each iteration hk, a general line search algorithm will:

1. choose a line (a.k.a. direction) along which to step

2. search along this line for a suitable step distance

For the specific Gradient Descent algorithm, the line along which to step is chosen

as the line of maximum negative rate of change (steepest descent). From the theory

of directional derivatives [149], this line is represented mathematically as the unit

vector:

u = −g∥g∥ (A.1)

where the Gradient vector g is defined as:

g ≜ ∂B(h)∂h

∣hk

=⎡⎢⎢⎢⎢⎣

∂B(h)∂h1

∂B(h)∂h2

⋯ ∂B(h)∂hn

⎤⎥⎥⎥⎥⎦

T ???????????hk

(A.2)

When B(h) has no closed form, g must be estimated numerically via the Finite

Differences or Least Squares techniques outlined in Appendix B.1.

To reap the maximum reward from this line of step u, the ideal step distance

µideal is computed as:

µideal = argminl >0

B(hk + lu) (A.3)

Solving (A.3) is no trivial task however, generally requiring an excessive number

of objective function measurements and gradient estimates. Since computational

170

APPENDIX A. SHORTLISTED OPTIMIZATION ALGORITHMS 171

complexity is undesirable in any optimization algorithm, practical implementations

of the Gradient Descent algorithm generally trade the ideal step of (A.3) for a

nonideal step µ requiring fewer computations.

While many computationally efficient methods exist for generating a µ value,

they all share the common trait of not computing µ directly, instead selecting µ

from a set of candidate steps [200]. The simple but reliable method used in this

research’s implementation of the Gradient Descent algorithm is outlined as follows:

• Let µhkrepresent the step distance chosen for iteration hk and cihk

represent

the ith candidate step distance generated during iteration hk.

• For iteration hk, candidate steps are generated as:

cihk=⎧⎪⎪⎪⎨⎪⎪⎪⎩

1.05µhk−1 i = 1

0.95 ci−1hki ≥ 2

(A.4)

Starting from c1hk, the first candidate step that brings about a reduction in

the objective is chosen as µhk. It is noted that for the very first iteration h1,

no previous step µh0 exists for use in (A.4), in which case µh0 is chosen based

on a priori knowledge.

• Four points are worth noting about this method for generating a step distance:

1. Since c1hk> µhk−1 , the step length is capable of growing.

2. Since cihk< µhk−1 for i ≥ 2, the step length is capable of shrinking.

3. Since cihk< ci−1hk

, the method is guaranteed to identify a step length which

reduces the objective function and is hence considered reliable.

4. Since µhkis chosen as the first candidate step bringing about a reduc-

tion in the objective function, subsequent better candidate steps could

potentially be missed. This is quite simply the tradeoff accepted for the

method’s computational efficiency.

A theoretically automated exit point from the Gradient Descent algorithm can-

not be defined based on first order objective function characteristics alone. While

a local minimum exhibits ∥g∥ = 0, so too does an objective function maximum or

saddle point. If at a particular iteration ∥g∥ = 0, differentiation between the three

possible cases requires computation of second order objective function characteris-

tics in the form of a Hessian matrix. A local minimum is subsequently identified

if the Hessian matrix is positive-definite. This being said however, if the objective

function has no closed form, estimation of this Hessian matrix is computationally

intensive and therefore generally avoided. In such cases, exiting from the Gradi-

ent Descent algorithm is generally left to the user’s discretion, for example, when


the algorithm continually fails to provide a meaningful objective function reduction.

This Gradient Descent algorithm is summarized in the flowchart of Figure A.1.

It is also implemented on the laboratory testbed via the GradientDescent() func-

tion. Corresponding declaration and definition source code resides in project files

GradientDescentOptimization.h andGradientDescentOptimization Templates.cpp re-

spectively; both files located within folder Software\Cpp\ on the accompanying DVD.

START

get current iterate

measure objective functionat current iterate

estimate Gradient vector

Minima, maxima or saddlepoint found!Not enough info to determine which.

Add small offset to to allowoptimization to continue.

compute unit vector in thedirection of maximum descent

compute candidate step distance

?

accept candidate step distance

step to

‘USER EXIT’request?

END

= 0 ?

set

increment

derive objective functionch

aracteristicscom

pute line along w

hich to step

compu

te step distance

compute another step ofoptimization algorithm

yes

no

no

no

yes

yes

g

∥g∥

cihk

cihk

(hk + µhku)

µhk= cihk

B(hk + cihku) < bo

g

u = −g∥g∥

hk

bo

cihk=⎧⎪⎪⎨⎪⎪⎩

1.05µhk−1 i = 1

0.95 ci−1hki ≥ 2

i

i = 1

Figure A.1: Flowchart of Gradient Descent optimization algorithm


A.2 Trust Region Newton

At each iteration hk of the Trust Region Newton (TRN) optimization algorithm,

the objective function is approximated by the second order Taylor series:

B(hk + d ) ≈ B(hk) + dT⎛⎝∂B(h)∂h

∣hk

⎞⎠+ 1

2dT⎛

⎝∂

∂h(∂B(h)

∂h)T

∣hk

⎞⎠d (A.5)

The right hand side of (A.5) thus represents a quadratic model of the objective

function at the current iteration hk:

m(d ) ≜ B(hk) + dT⎛⎝∂B(h)∂h

∣hk

⎞⎠+ 1

2dT⎛

⎝∂

∂h(∂B(h)

∂h)T

∣hk

⎞⎠d (A.6)

It can be seen from (A.6) that at the current iterate, the model and objective func-

tion characteristics are identical up to second order. That is:

m(d )∣d=0 = B(hk) (A.7)

∂m(d )∂d

∣d=0

=⎛⎝∂B(h)∂h

∣hk

⎞⎠+⎛⎝

∂

∂h(∂B(h)

∂h)T

∣hk

⎞⎠d

???????????d=0= ∂B(h)

∂h∣hk

(A.8)

∂

∂d(∂m(d )

∂d)T

∣d=0

=⎛⎝

∂

∂h(∂B(h)

∂h)T

∣hk

⎞⎠

???????????d=0= ∂

∂h(∂B(h)

∂h)T

∣hk

(A.9)

To simplify notation, the (A.6) model can be rewritten as:

m(d ) ≜ bo + dTg + 1

2dTHd (A.10)


where

bo ≜ B(hk) (A.11)

g ≜ ∂B(h)∂h

∣hk

=⎡⎢⎢⎢⎢⎣

∂B(h)∂h1

∂B(h)∂h2

⋯ ∂B(h)∂hn

⎤⎥⎥⎥⎥⎦

T ???????????hk

(A.12)

H ≜ ∂

∂h(∂B(h)

∂h)T

∣hk

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂2B(h)∂h1 ∂h1

∂2B(h)∂h1 ∂h2

⋯ ∂2B(h)∂h1 ∂hn

∂2B(h)∂h2 ∂h1

∂2B(h)∂h2 ∂h2

⋯ ∂2B(h)∂h2 ∂hn

⋮ ⋮ ⋱ ⋮

∂2B(h)∂hn ∂h1

∂2B(h)∂hn ∂h2

⋯ ∂2B(h)∂hn ∂hn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

???????????????????????????????????????????????????hk

(A.13)

g and H are referred to as the Gradient vector and Hessian matrix respectively.

When the objective function B(h) has no closed form, g and H must be estimated

numerically via the techniques of Appendix B.1 and B.2 respectively. H may also

be estimated via Symmetric-Rank-1 Updating, as discussed later in this section,

for all iterations other than the very first. In all such cases where g and H are

estimated numerically, the optimization algorithm is technically referred to as the

Trust Region Quasi -Newton algorithm. It is noted that both g and H are real and

H is symmetric.

In order to quantify the domain of this model m(d ), an accompanying trust

region radius ∆ > 0 must be specified. A trust region is defined as a spherical region

(ball of radius ∆) around the current iterate within which the quadratic modelm(d )is trusted to be an accurate representation of the objective function B(hk + d ).

When both the quadratic model m(d ) and its accompanying trust region ra-

dius ∆ are estimated at the current iteration hk, a candidate optimization step is

computed by minimizing the model within the current trust region:

mind

m(d ) = bo + dTg + 1

2dTHd subject to ∥d ∥ ≤ ∆ (A.14)

This minimizer d∗ represents the candidate optimization step. Ifm(d∗ ) andB(hk + d∗ )are comparable, the candidate step is locked in. If however m(d∗ ) and B(hk +d∗ )


are significantly different, the current ∆ estimate is considered to be over-estimated.

In this case, the ∆ estimate must be reduced and the minimization of (A.14) re-

computed. In general, the candidate optimization step will be different for each ∆

estimate.

The size of the trust region radius estimate is crucial in the effectiveness of this

algorithm. If the radius is under-estimated, the algorithm will miss the opportunity

to take a larger step towards the objective function minimum and the algorithm

will take longer to converge. If on the other hand the radius is over-estimated, the

model won’t adequately represent the objective function over the entire region and

the candidate step will in general be misleading. In practice, the initial trust region

radius estimate at iteration hk is chosen to be slightly larger than the final estimate

of the previous iteration hk−1. This allows the trust region radius to grow, thus

avoiding potential future under-estimation and slow convergence, at the same time

keeping the number of reduction re-estimates at each iteration to a minimum.

[185] states that the solution d∗ of (A.14) must satisfy the following conditions

for some scalar λ ≥ 0:

[H + λI]d∗ = −g (A.15a)

λ (∆ − ∥d∗∥) = 0 (A.15b)

H + λI is positive semidefinite (A.15c)

These conditions suggest a two case strategy for finding the solution d∗ [200]:

Case 1: If λ = 0 satisfies (A.15c) (specifically H is positive-definite and therefore

nonsingular) and the solution d∗ = −H−1g of (A.15a) is within the trust region

(∥d∗∥ ≤ ∆) then d∗ represents the solution of (A.14). This solution is known

as the unconstrained minimum of the model m(d ).

Case 2: If Case 1 does not hold, the solution d∗ of (A.15a) must be evaluated as a

function of λ:

d∗(λ) = − [H + λI]−1 g (A.16)

specifically for λ satisfying (A.15c) and the solution of (A.14) must be chosen

as that d∗(λ) for which ∥d∗ (λ)∥ = ∆, thus satisfying (A.15b).

Before continuing, the reader is encouraged to consult Appendix B.3 to gain

familiarity with the eigen properties of H . Knowledge of these properties will be

assumed in the following discussion.


Evaluating The Solution Via Case 1

The first step in evaluating the solution of (A.14) via Case 1 is to determine whether

H is positive-definite or not. H is positive-definite if all of its eigenvalues are positive

(Property 4, Appendix B.3). The eigenvalues of H are obtained by decomposing H

into its diagonal form H = QΛQT via the QR-Factorization method [88] and strip-

ping off the diagonal elements of the spectral matrix Λ (Property 2, Appendix B.3).

If these eigenvalues are all positive and H is subsequently positive-definite, the

unconstrained minimum solution d∗ = −H−1g of (A.15a) must be evaluated. Rather

than computing a matrix inverse however, Hd∗ = −g is treated as a linear system

of equations and solved efficiently via Cholesky decomposition [143].

Once the above unconstrained minimum d∗ has been computed, its position with

respect to the trust region must be determined. If:

∥d∗∥ = (d∗Td∗)12 ≤ ∆ (A.17)

the unconstrained minimum lies within the trust region and therefore represents the

solution of (A.14) via Case 1.

Evaluating The Solution Via Case 2

If H is not positive-definite or the unconstrained minimum is not within the trust

region, the solution of (A.14) must be evaluated via Case 2. In this case, the solution

d∗ of (A.15a) must be evaluated as a function of λ:

d∗(λ) = − [H + λI]−1 g (A.18)

specifically for λ satisfying (A.15c) and the solution of (A.14) must be chosen as

that d∗(λ) for which ∥d∗(λ)∥ = ∆, thus satisfying (A.15b).

Decomposing [H + λI] into its diagonal form Q [Λ + λI]QT (Property 2 & 5,

Appendix B.3) and substituting into (A.18) gives:

d∗(λ) = −Q [Λ + λI]−1QTg (A.19)

= −n

∑i=1

( qTi g

λi + λ)qi (A.20)

where qi represents the ith eigenvector of H and therefore the ith column of Q.

Given the orthogonality of Q, ∥d∗(λ)∥ can be expressed as:

∥d∗(λ)∥ =⎛⎝

n

∑i=1

( qTi g

λi + λ)2 ⎞⎠

12

(A.21)


As stated above, the goal now is to find that value of λ satisfying (A.15c), for

which ∥d∗(λ)∥ = ∆. The corresponding d∗(λ) subsequently represents the solution

of (A.14). From (A.21), it can be seen that two scenarios exist in this search for λ.

Case 2 - Scenario 1 When qT1 g ≠ 0, ∥d∗(λ)∥ in (A.21) is a continuous, nonin-

creasing function of λ ∈ (−λ1,∞) for which:

limλ→−λ1

∥d∗(λ)∥ = ∞ and limλ→∞

∥d∗(λ)∥ = 0 (A.22)

In this scenario, a unique λ ∈ (−λ1,∞) exists for which ∥d∗(λ)∥ = ∆. This λ is

computed via Newton’s root-finding method [200] as follows.

On the open interval λ ∈ (−λ1,∞), the function:

φ(λ) = ∥d∗(λ)∥ −∆ (A.23)

will have a root at the unique value of λ for which ∥d∗(λ)∥ = ∆. Given an ini-

tial estimate of the root λ0 ∈ (−λ1,∞), the iterative Newton’s root-finding method

estimates an improved estimate according to:

λk+1 = λk − φ(λk)φ′(λk) where φ′(λk) = dφ

dλ∣λk

(A.24)

This iteration continues until φ(λk+1) approaches zero at which point λk+1 represents

the desired value of λ and the corresponding d∗(λ) represents the solution of (A.14).

In practice, safeguards are put in place to ensure λk+1 remains greater than −λ1

during the iteration process.

The performance of the Newton’s root-finding method improves with the linear-

ity of the function φ(λ). For this reason, the alternative, more linear function:

φ(λ) = 1

∆− 1

∥d∗(λ)∥ (A.25)

is preferred over (A.23) which is significantly nonlinear when λ is greater than but

close to −λ1. If (A.25) is indeed utilized, [185] shows that (A.24) can be computed

efficiently as:

λk+1 = λk + (∥dk∥

∥qk∥)2

(∥dk∥ −∆

∆) (A.26)


where dk and qk are computed according to:

H + λkI = RTR (A.27a)

RTRdk = −g (A.27b)

RTqk = dk (A.27c)

Here, R represents the upper triangular Cholesky factorization matrix.

Case 2 - Scenario 2 When the current iteration hk is positioned on one of the

principle axes defined by eigenvectors q2 → qn and consequently qT1 g = 0 (Property 3,

Appendix B.3), ∥d∗(λ)∥ in (A.21) is once again a continuous, nonincreasing function

of λ ∈ (−λ1,∞) but this time:

limλ→−λ1

∥d∗(λ)∥ =⎛⎝

n

∑i=2

( qTi g

λi − λ1)2 ⎞⎠

12

and limλ→∞

∥d∗(λ)∥ = 0 (A.28)

If the trust region radius ∆ is such that:

∆ < limλ→−λ1

∥d∗(λ)∥ (A.29)

then a unique λ ∈ (−λ1,∞) exists for which ∥d∗(λ)∥ = ∆ and the corresponding

d∗(λ) represents the solution of (A.14). This λ is computed via Newton’s root-

finding method in exactly the same manner as in Scenario 1 previously.

If however the trust region radius ∆ is such that:

∆ ≥ limλ→−λ1

∥d∗(λ)∥ (A.30)

then a unique value λ ∈ (−λ1,∞) will not exist for which ∥d∗(λ)∥ = ∆. According

to (A.15c), in this rare case λ must take on the value −λ1. For λ = −λ1, [H − λ1I]is singular (at least one eigenvalue equals zero and therefore the determinant equals

zero (Property 6, Appendix B.3)) and (A.15a) has an infinite number of solutions

of the form:

d∗ =⎛⎝−

n

∑i=2

( qTi g

λi − λ1)qi

⎞⎠+ τq1 where −∞ < τ < ∞ (A.31)

From all of these solutions, the goal is to choose that unique solution d∗ for which

∥d∗∥ = ∆. Taking the norm of (A.31) and equating to ∆ gives:

∥d∗∥ =⎡⎢⎢⎢⎢⎣

⎛⎝

n

∑i=2

( qTi g

λi − λ1)2 ⎞⎠+ τ2

⎤⎥⎥⎥⎥⎦

12

= ∆ (A.32)


Rearranging (A.32) then produces the expression for τ :

τ =⎡⎢⎢⎢⎢⎣∆2 −

⎛⎝

n

∑i=2

( qTi g

λi − λ1)2 ⎞⎠

⎤⎥⎥⎥⎥⎦

12

(A.33)

corresponding to the unique solution d∗ of (A.31) for which ∥d∗∥ = ∆. This d∗

subsequently represents the solution of (A.14). To be precise, it is noted that two

solutions d∗ actually exist for which ∥d∗∥ = ∆. These solutions corresponding to

τ being positive and negative in (A.33). As a standard operating procedure, the

solution corresponding to positive τ is chosen. It is worth noting that [185] refers to

this Case 2 - Scenario 2 as the hard case and the left hand limit of (A.28) as the

limiting trust region radius.

Estimating H Via Symmetric-Rank-1 Updating

For the very first iteration of the Trust Region Quasi -Newton algorithm, H must be

estimated via the techniques of Appendix B.2. For all subsequent iterations however,

an additional technique for estimating H becomes available, this technique being

Symmetric-Rank-1 Updating.

Let Hk and Hk−1 represent the Hessian matrices at iterations hk and hk−1

respectively. Similarly, let gk and gk−1 represent the Gradient vectors at itera-

tions hk and hk−1 respectively. Instead of estimating Hk afresh at iteration hk,

the Symmetric-Rank-1 Updating technique estimates Hk from the already avail-

able Hk−1, gk−1 and gk estimates, making it significantly more efficient than the

techniques of Appendix B.2. This efficiency, coupled with good estimation accu-

racy [200], makes the technique very popular in practical implementations of the

Trust Region Quasi -Newton algorithm.

In the Symmetric-Rank-1 Updating technique, the estimate of Hk must satisfy

two conditions [74]:

Condition 1

Hk (hk −hk−1) = gk − gk−1 (A.34)

This is called the Secant equation and ensures that Hk is estimated such that:

∂mk(d )∂d

∣d=−(hk−hk−1)

= gk +Hkd ∣d=−(hk−hk−1)

≡ gk−1 (A.35)

where mk(d ) is the quadratic objective function model (A.10) at iteration hk.

Put simply, this condition ensures that the Gradient of mk(d ) matches the

Gradient of the objective function at the current and previous iteration.


Condition 2

Hk = Hk−1 + σ vvT (A.36)

This equation represents the general form of Hk where σ (+1 or −1) and v

(n element column vector) are computed such that the Secant equation of

Condition 1 is satisfied. It is called the Symmetric-Rank-1 Update equation

since the outer product vvT is symmetric with unity rank [11] and Hk is

obtained by updating the previous Hk−1. It is now obvious as to the naming

origins of this Hessian estimation technique.

Substituting the Symmetric-Rank-1 Update equation into the Secant equation gives:

(Hk−1 + σ vvT )(hk −hk−1) = gk − gk−1 (A.37)

To simplify notation, let:

sk = hk −hk−1 (A.38)

yk = gk − gk−1 (A.39)

(A.37) can then be rewritten compactly as:

(Hk−1 + σ vvT )sk = yk (A.40)

Rearranging (A.40) then gives:

[σ vTsk]v = yk −Hk−1sk (A.41)

Since the square bracketed term is a scalar, it logically follows that v must be a

multiple of (yk −Hk−1sk). In this context, let v take the general form:

v = ϕ(yk −Hk−1sk) for some scalar ϕ (A.42)

Substituting this general form of v back into (A.41) then gives:

σϕ2sTk (yk −Hk−1sk)(yk −Hk−1sk) = yk −Hk−1sk (A.43)

It is clear that (A.43) is satisfied if and only if σ and ϕ are chosen as:

σ = sign ( sTk (yk −Hk−1sk) ) (A.44)

ϕ = ± ∣sTk (yk −Hk−1sk) ∣− 1

2(A.45)

Substituting (A.44), (A.45) and (A.42) into (A.36) then gives the desired Symmetric-


Rank-1 Update estimate of Hk:

Hk = Hk−1 +(yk −Hk−1sk)(yk −Hk−1sk)

T

sTk (yk −Hk−1sk)(A.46)

where sk = hk−hk−1 and yk = gk−gk−1. As mentioned previously, since this estimate

of Hk is based on the already available Hk−1, gk−1 and gk estimates, it is computa-

tionally efficient. This efficiency, coupled with good estimation accuracy, makes the

technique popular in practical implementations of the Trust Region Quasi -Newton

algorithm. In the rare case when the denominator term alone approaches 0, prac-

tical implementations simply skip the update and set Hk = Hk−1 with negligible

negative effects being observed.

It is worth noting that other Hessian updating techniques exist, for example the

BFGS and DFP methods [51], however these are only appropriate for convex ob-

jective functions and hence aren’t suitable for the Trust Region Newton algorithm

discussed here.

This Trust Region Newton algorithm is summarized in the flowchart of Fig-

ure A.2. It is also implemented on the laboratory transmitter testbed via the

software template function TrustRegionNewton(). Corresponding declaration and

definition source code resides in project files TrustRegionNewtonOptimization.h and

TrustRegionNewtonOptimization Templates.cpp respectively. Both files are located

within folder Software\Cpp\ on the accompanying DVD.

START

estimate Gradient vector

estimate Hessian matrix

derive objective function model

perform eigendecomposition of

positive-definite?

minima found!END

compute unconstrainedminimum of

Maxima or saddlepoint found!Add small offset to to

allow optimization tocontinue towards minima.

Hard Caseidentified? compute limiting

|| unconstrained min || ?

candidate stepequals

unconstrainedminimum

perform Newton’s rootfinding method to

compute (note: )

candidate step equals

compute corresponding to

compute candidatestep corresponding

to

candidate step valid?Reduce < || candidate step ||and recompute candidate step

based on new

Move to candidate step! In preparation for next step,set > || candidate step ||. Compute another step of

optimization algorithm.

= 0 ?

yes

no

yesyes

no

no

no

yes

analyse and compute characteristics of

measure objective functionat current iterate

get current iterate

= 0 ?

positive-definite?

Is Hard Case AND limiting ?

yesno noyes

perform Newton’s rootfinding method to

compute (note: )

candidate step equals

yesno

compute candidate step

no

yes no

Case 1 Case 2 Case 2 Case 2

λλλ > 0

τ

τ

λ > −λ1

∆ ≥ ∆

∆∆

∆

∆

∆

≤∆

g

g

H

H

H

H

hk

bo

m(d)

≜bo +

dTg

+12dTH

d

∥g∥∥g∥

m(d)

m(d)

− [H + λI]−1 g−[H + λI]−1 g

Figure A.2: Flowchart of Trust Region Newton optimization algorithm


A.3 Alpha Branch & Bound

The Alpha Branch & Bound (ABB) optimization algorithm is gradient based with

global scope [12]. The algorithm consists of three phases as outlined below.

Phase 1

1. The vector space h is partitioned into boxed regions.

2. Lower & upper bounds on each region’s objective function minimum are esti-

mated, a process referred to as bounding. It is worth noting that any future

reference to region lower and upper bounds is made in this context unless

otherwise stated.

3. From the upper bounds of all regions, the overall minimum upper bound is

identified. Any region whose lower bound is greater than this overall minimum

upper bound is discarded, a process referred to as fathoming.

Phase 2

1. From all of the unfathomed regions, the region with the largest bound interval

(upper bound subtract lower bound) is identified. This region is bisected along

its longest side to give two smaller regions, a process referred to as branching.

2. Lower & upper bounds on each new region’s objective function minimum are

estimated (bounding). The two new regions are added to the list of unfathomed

regions whilst the original bisected region is discarded.

3. From the upper bounds of all unfathomed regions, the overall minimum upper

bound is identified. Any unfathomed region whose lower bound is greater than

this overall minimum upper bound is discarded (fathoming).

4. Phase 2 is repeated. With smaller regions comes tighter bound intervals and

refined region fathoming.

After numerous iterations of Phase 2 (in accordance with Step 4), the number and

geometric size of unfathomed regions dramatically reduces. These unfathomed re-

gions subsequently represent candidate regions within which the objective function

global minimum could reside. Repetition of Phase 2 ends when the number of un-

fathomed regions drops to less than some small predefined number κ chosen by the

user.


Phase 3

1. For each remaining unfathomed region, a local optimizer is employed to locate

the region’s objective function minimum. Only one minimum is assumed in

each unfathomed region given its small geometric size.

2. Based on all of the minima identified in Step 1, the smallest is considered the

objective function global minimum over the entire vector space.

The concepts of Phases 1 – 3 are discussed in further detail in the following. Be-

fore continuing however, the reader is encouraged to consult Appendix B.3 to gain

familiarity with the eigen properties of the objective function Hessian matrix H .

Knowledge of these properties will be assumed in the following discussion.

Partitioning The Vector Space Into Boxed Regions

Despite the vector space h ∈ Rn being theoretically unconstrained, a practical lower

constraint hL and upper constraint hU is set in order to establish a finite search

space:

hL ≤ h ≤ hU (A.47)

By dividing each dimension 1 ≤ i ≤ n of the search space into K equal intervals:

hUi − hLiK

(A.48)

the search space can be partitioned into Kn boxed regions, each with a domain of

the form:

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

hL1 + k1 (hU1 − hL1

K) ≤ h1 ≤ hL1 + (k1 + 1) (h

U1 − hL1K

)

hL2 + k2 (hU2 − hL2

K) ≤ h2 ≤ hL2 + (k2 + 1) (h

U2 − hL2K

)

⋮ ⋮ ⋮

hLn + kn (hUn − hLn

K) ≤ hn ≤ hLn + (kn + 1)(h

Un − hLnK

)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

for integers 0 ≤ ki < K

(A.49)

Since the domain of each boxed region is dependent on the vector:

k = [k1, k2,⋯, kn] for integers 0 ≤ ki < K (A.50)


each region can be uniquely identified as Rk with domain:

[hL,Rk ≤ h ≤ hU,Rk] (A.51)

where the lower constraint hL,Rk and upper constraint hU,Rk represent the left and

right hand side respectively of (A.49) for the specific k vector:

hL,Rk =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

hL1 + k1 (hU1 − hL1

K)

hL2 + k2 (hU2 − hL2

K)

⋮

hLn + kn (hUn − hLn

K)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

hU,Rk =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

hL1 + (k1 + 1)(hU1 − hL1K

)

hL2 + (k2 + 1)(hU2 − hL2K

)

⋮

hLn + (kn + 1)(hUn − hLnK

)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A.52)

In order to keep the initial number of boxed regions to a practical limit, knowing

that each region needs to be bounded in Phase 1 Step 2, K is chosen reasonably

small. In this sense, the idea is to partition the search space fairly coarsely and then

let the algorithm in Phase 2 determine which regions to further branch and bound.

It is worth noting that, as a consequence of the algorithm’s repeated Phase 2

Step 2, these partitioned regions Rk will be progressively bisected into smaller boxed

regions. In general, these smaller boxed regions do not take the specific form of

(A.49) and therefore cannot be uniquely identified by the vector subscript k. For

this reason, the remaining discussion will be in terms of the general boxed region

Rg whose domain is of the form:

[hL,Rg ≤ h ≤ hU,Rg] =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

hL,Rg

1 ≤ h1 ≤ hU,Rg

1

hL,Rg

2 ≤ h2 ≤ hU,Rg

2

⋮ ⋮ ⋮

hL,Rgn ≤ hn ≤ h

U,Rgn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A.53)

Since the region resides within the finite search space, its lower constraint hL,Rg and

upper constraint hU,Rg satisfies:

hL ≤ hL,Rg ≤ hU,Rg ≤ hU (A.54)


Bounding

Given a general boxed region Rg, bounding is the process of estimating a lower and

upper bound on that region’s objective function minimum:

minhL,Rg ≤h≤hU,Rg

B(h) (A.55)

When estimating a lower bound on the region’s objective function minimum, the idea

is to engineer a convex function L(h) which under-estimates the objective function

B(h) over the entire domain of the region. Convexity ensures that L(h) has a

single minimum within the region’s domain and under-estimation ensures that this

minimum is less than the region’s objective function minimum, thereby representing

a true lower bound. A local optimization algorithm is employed to locate the single

minimum of L(h) within the region.

Let Φ be defined as the diagonal matrix:

Φ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

α1 0 ⋯ 0

0 α2 ⋯ 0

⋮ ⋮ ⋱ ⋮0 0 ⋯ αn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

where αi ≥ 0 (A.56)

The convex under-estimating function L(h) then takes the general form [4]:

L(h) = B(h) + (hL,Rg −h)TΦ (hU,Rg −h) (A.57a)

= B(h) +n

∑i=1

αi (hL,Rg

i − hi)(hU,Rg

i − hi) (A.57b)

Four points are worth noting about (A.57):

1. With αi ≥ 0 and [hL,Rg ≤ h ≤ hU,Rg], the right hand summation term of (A.57b)

is seen to be nonpositive over the entire region, hence ensuring L(h) is a valid

under-estimator of B(h). L(h) matches B(h) at all region corner points.

2. The right hand quadratic term of (A.57a) can be interpreted geometrically as

the convex quadratic surface θ (h) = hTΦh translated to the region’s centroid

(hU,Rg +hL,Rg) /2 and then de-elevated to ensure negativity over the entire


region. That is:

(hL,Rg −h)TΦ (hU,Rg −h) = θ⎛⎝h − [h

U,Rg +hL,Rg

2]⎞⎠

− θ⎛⎝hL,Rg − [h

U,Rg +hL,Rg

2]⎞⎠

(A.58)

In this sense, the Hessian matrix of the right hand quadratic term of (A.57a)

is simply the Hessian matrix of θ (h) = hTΦh = (12)hT (2Φ) h which is 2Φ.

It logically follows from (A.57a) and the linearity property of derivatives that

the Hessian matrix of L(h) is given by:

HL(h) = HB(h) + 2Φ (A.59)

where HB(h) represents the Hessian matrix of the objective function B(h).

3. The values of αi are chosen specifically to ensure regional convexity of L(h). Toachieve regional convexity, HL(h) must be positive-definite and therefore pos-

sess all positive eigenvalues (Property 4, Appendix B.3) for [hL,Rg ≤ h ≤ hU,Rg].

From (A.59) and Gerschgorin’s Theorem (Property 8, Appendix B.3), HL(h)’sith eigenvalue λi(h) is bounded by:

⎛⎜⎜⎝HB i,i(h) −

n

∑j=1j≠i

∣HB i,j(h)∣⎞⎟⎟⎠+ 2αi ≤ λi(h) ≤

⎛⎜⎜⎝HB i,i(h) +

n

∑j=1j≠i

∣HB i,j(h)∣⎞⎟⎟⎠+ 2αi

(A.60)

whereHB i,j(h) is the (i, j)th element of HB(h). It is noted that in (A.60), the

subscript i of λi(h) signifies the Gerschgorin disk Di with which the eigenvalue

is associated. This is not to be confused with the eigenvalue subscript used in

Properties 1 and 2 of Appendix B.3 which denotes the relative magnitude of

eigenvalues.

To ensure the lower bound of (A.60) is positive for all [hL,Rg ≤ h ≤ hU,Rg],thereby ensure positive eigenvalues and hence regional convexity of L(h),αi ≥ 0 must satisfy:

αi ≥ max

⎧⎪⎪⎨⎪⎪⎩0, −1

2

⎛⎝


(HB i,i(h) −n

∑j=1j≠i

∣HB i,j(h)∣ )⎞⎠

⎫⎪⎪⎬⎪⎪⎭(A.61)

The fact that αi is bounded below according to (A.61) can be understood

intuitively by recognizing that the αi’s determine the convexity of the right


hand quadratic term of (A.57a). By choosing the αi’s sufficiently large, the

convexity of this quadratic term can be made to overpower all nonconvexities

in the objective function B(h) and hence guarantee convexity of L(h).

4. It is shown in [10,175] that L(h) under-estimates the objective function B(h)by a maximum:

maxhL,Rg ≤h≤hU,Rg

(B(h) − L(h)) = 1

4

n

∑i=1

αi (hU,Rg

i − hL,Rg

i )2

(A.62)

It follows that minimal αi values and smaller boxed regions lead to tighter

under-estimating functions. This in general leads to tighter bound intervals

(upper bound subtract lower bound) and hence refined region fathoming.

Based on Dot-Points 3 and 4 above, the ideal value of αi, that which leads to the

tightest convex under-estimating function, is given by:

αideali = max

⎧⎪⎪⎨⎪⎪⎩0, −1

2

⎛⎝


(HB i,i(h) −n

∑j=1j≠i

∣HB i,j(h)∣ )⎞⎠

⎫⎪⎪⎬⎪⎪⎭(A.63)

Unfortunately, unless HB(h) is strictly analytic, the right hand minimization term

of (A.63) is virtually impossible to compute, leaving αideali unobtainable in practice.

In such situations, this problematic right hand minimization term is completely

replaced with its practically computable lower bound to give, by definition, a gen-

erally nonideal though valid value of αi, subsequently referred to as αpraci to denote

its practical computability. The lower bound on the right hand minimization term

is computed along with αpraci as follows:

• The region’s objective function interval Hessian matrix [HB]Rg = [¯HB ,HB ]

(Property 7, Appendix B.3) is estimated from numerous samples of HB(h)taken within the region [hL,Rg ≤ h ≤ hU,Rg]. These Hessian samples are them-

selves estimated via the techniques of Appendix B.2. During the interval esti-

mation process, elements of theHB(h) samples are treated independently [197,

224]. A lower bound on the ith eigenvalue of { [HB]Rg} is then computed via

the principles of Gerschgorin’s Theorem as:

¯HB i,i −

n

∑j=1j≠i

max ( ∣¯HB i,j ∣, ∣HB i,j ∣ ) (A.64)

Here¯HB i,j and HB i,j represent the (i, j)th elements of

¯HB and HB respectively.


• With Gerschgorin’s Theorem still in mind, the right hand minimization term

of Equation (A.63) is recognized as the lower bound on the ith eigenvalue of

{HB(h) ∀ [hL,Rg ≤ h ≤ hU,Rg] }. Since by definition:

{HB(h) ∀ [hL,Rg ≤ h ≤ hU,Rg] } ⊆ { [HB]Rg} (A.65)

it follows that:

¯HB i,i −

n

∑j=1j≠i

max ( ∣¯HB i,j ∣, ∣HB i,j ∣ )

≤ minhL,Rg ≤h≤hU,Rg

(HB i,i(h) −n

∑j=1j≠i

∣HB i,j(h)∣ ) (A.66)

and (A.64) represents the sought after lower bound on the right hand mini-

mization term of (A.63).

• The right hand minimization term of (A.63) is then completely replaced with

this (A.64) lower bound to give the practically computable αpraci :

αpraci = max

⎧⎪⎪⎨⎪⎪⎩0, −1

2

⎛⎝¯HB i,i −

n

∑j=1j≠i

max ( ∣¯HB i,j ∣, ∣HB i,j ∣ )

⎞⎠

⎫⎪⎪⎬⎪⎪⎭(A.67)

By definition, αpraci ≥ αideal

i and hence convexity condition (A.61) is satisfied.

With αi computed via (A.67), the general form of L(h) originally expressed in (A.57)

is now fully defined. As discussed previously, since L(h) is a convex under-estimator

of the objective function B(h) over the entire region Rg, its single regional minimum

represents the sought after lower bound on the region’s objective function minimum.

L(h)’s single regional minimum is located via a local optimization algorithm oper-

ating on (A.57).

A valid upper bound on the region’s objective function minimum is estimated

simply as the value of the objective function B(h) at L(h)’s single regional mini-

mum.

Fathoming

Given a set of general boxed regions and the corresponding lower and upper bounds

on each region’s objective function minimum, fathoming is the process of identifying

and discarding regions which cannot theoretically contain the objective function’s

global minimum. The identification process is based on comparison of each region’s

lower bound to the overall minimum upper bound.


boun

ds o

n ea

ch r

egio

n’s

obje

ctiv

e fu

nctio

n m

inim

um

regionRA RB RC RD RE

Figure A.3: Fathoming example for the Alpha Branch & Bound algorithm

For example, consider the set of general boxed regions {RA, RB , RC , RD, RE }and let the lower and upper bounds on each region’s objective function minimum be

represented graphically in Figure A.3. Here, ▽ and △ represent the lower and upper

bounds respectively for each region whilst the continuous line connecting ▽ and △represents the corresponding bound interval. From the upper bounds of all regions

in Figure A.3, the overall minimum upper bound is seen to belong to RE . Since the

lower bound of RB is greater than the upper bound of RE , it logically follows that

RB’s objective function minimum must be theoretically greater than RE ’s objective

function minimum and therefore RB cannot possibly contain the global objective

function minimum. In this case, RB can be discarded from further analysis. RB is

then said to be fathomed.

At this point in the example, nothing more can be ascertained about the location

of the global objective function minimum from the remaining unfathomed set of

region’s and bounds and therefore branching must take place.

Branching

Given a set of unfathomed general boxed regions and the corresponding lower and

upper bounds on each region’s objective function minimum, branching is the process

of identifying the region with the largest bound interval (upper bound subtract lower

bound) and bisecting this region along its longest side to give two smaller regions.

Continuing on from the previous fathoming example, the following four unfath-

omed regions remain {RA, RC , RD, RE }. The lower and upper bounds on each

region’s objective function minimum are taken from Figure A.3 and repeated in


boun

ds o

n ea

ch r

egio

n’s

obje

ctiv

e fu

nctio

n m

inim

um

regionRA RC RD RE

Figure A.4: Branching example for the Alpha Branch & Bound algorithm

Figure A.4. Once again, ▽ and △ represent the lower and upper bounds respec-

tively for each region whilst the continuous line connecting ▽ and △ represents the

corresponding bound interval. From the bound intervals of all regions in Figure A.4,

the largest bound interval is seen to belong to RC . Let the domain of region RC be

represented by:

[hL,RC ≤ h ≤ hU,RC] =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

hL,RC1 ≤ h1 ≤ hU,RC

1

⋮ ⋮ ⋮

hL,RC

l ≤ hl ≤ hU,RC

l

⋮ ⋮ ⋮

hL,RCn ≤ hn ≤ hU,RC

n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A.68)

where hU,RC

l − hL,RC

l represents the region’s longest side, that is:

l = argmaxi

(hU,RCi − hL,RC

i ) (A.69)

Region RC is then bisected along its lth side to give two new regions:


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


1

⋮ ⋮ ⋮

hL,RC

l ≤ hl ≤(hU,RC

l + hL,RC

l )2

⋮ ⋮ ⋮


n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


1

⋮ ⋮ ⋮

(hU,RC

l + hL,RC

l )2

≤ hl ≤ hU,RC

l

⋮ ⋮ ⋮


n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A.70)

This Alpha Branch & Bound algorithm is summarized in the flowchart of Fig-

ure A.5. It is worth noting that the additional Phase 1 yellow steps are optional.

If the user has application-specific a priori knowledge that outer Rk corner regions

(those with a large centroid norm) don’t possess the objective function global min-

imum, then such regions can be discarded from the very outset, thereby reducing

the algorithm’s initial bounding load and hence increasing initial algorithm speed.

This algorithm is implemented on the laboratory transmitter testbed via the

software template function AlphaBranchBound(). Corresponding declaration and

definition source code resides in project files AlphaBranchBoundOptimization.h and

AlphaBranchBoundOptimization Templates.cpp respectively. Both files are located


START

partition vector space into boxed regions

choose a region

perform a pre-validity check of regionbased on its || centroid ||

region valid?

discard region

estimate region’s objective function interval

Hessian matrix

compute and based on

estimate lower & upper bounds on region’s objective function minimum based on

and

add region to initial region set

perform re

gion bounding and

add region

to initial region set

Perform fathoming of initial region set.Unfathomed regions now form current region set.

size( current region set ) > ?

perform branching to create two newgeneral boxed regions

for both new regions … estimate

the objective function interval

Hessian matrix

for both new regions … compute

and based on

for both new regions … estimate lower &upper bounds on region’s objective function minimum based on and

add both new regions to current region set

for both new reg

ions … perfo

rm region

bounding and add reg

ion to current reg

ion set

perform fathoming of current region set

for each remaining unfathomed region in thecurrent region set … locate region’s objective

function minimum via local optimization

END

from all the regional minima identified in theprevious step, the smallest is considered the

objective function global minimum!

last region?

Phase 1

Phase 2

Phase 3

yes

no

no yes

yes no

Note: yellow steps optional toincrease initial algorithm speed

h

Rk

Rk

[HB]Rk

[HB]Rk

αpraci

αpraci

κ

Rg

[HB]Rg

[HB]Rg

Φ

Φ

Φ

Φ

L(h)

L(h)

Figure A.5: Flowchart of Alpha Branch & Bound optimization algorithm


A.4 Nelder-Mead Simplex

Developed by J.A. Nelder and R. Mead [196] from the earlier work of [260], this

optimization algorithm uses basic simplex geometry plus objective function measure-

ment and comparison to identify a local objective function minimum. A simplex is

simply the convex hull of a set of (n + 1) points in the Rn vector space [111]. A

triangle for example is a simplex in the R2 space, likewise a tetrahedron is a simplex

in the R3 space. In its most basic form, the optimization algorithm involves:

• initially defining an (n + 1) vertex simplex in the Rn vector space about the

starting point of the optimization.

• updating the geometry of this simplex at each optimization step, via either

reflection , contraction, expansion or shrinkage, with the general goal of reduc-

ing its vertex objective function values. Viewed abstractly in terms of the local

objective function landscape, this geometric updating process sees the simplex

elongate down long inclined planes, change direction on encountering a valley

at an angle and contract in the neighborhood of a minimum [196].

In the context of the first Dot-Point above, let the initial simplex S0 be defined by

the set of (n + 1) vertices:

S0 =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

hs, hs +

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

d

0

⋮0

0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, hs +

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0

d

⋮0

0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, ⋯ , hs +

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0

0

⋮d

0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, hs +

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0

0

⋮0

d

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(A.71)

where hs is the vector space starting point of the mathematical optimization and d

is a small positive increment.

In the context of the second Dot-Point above, let Sk represent the geometrically

updated simplex at optimization step k (k ≥ 1). Sk is then computed as follows:

1. Vertices of the previous simplex Sk−1 are uniquely labeled according to their

relative objective function values. The vertex exhibiting the lowest objective

function value is labeled v1, the vertex exhibiting the second lowest objective

function value is labeled v2 and so on up to the vertex exhibiting the highest

objective function value which is labeled vn+1. That is:

B(v1) ≤ B(v2) ≤ ⋯ ≤ B(vn) ≤ B(vn+1) (A.72)


2. The centroid of Sk−1’s best n vertices is computed as:

v = ( 1n)

n

∑i=1

vi (A.73)

3. The reflection point of Sk−1 is defined and computed as:

vr = v − (vn+1 − v) (A.74)

This reflection point lies along the line intersecting the centroid v and the

worst vertex vn+1. B(vr) is then measured.

4. The desired Sk is formed by either reflecting, contracting, expanding or shrink-

ing Sk−1 based on the value of B(vr) relative to B(v1), B(vn) and B(vn+1):

If B(v1) ≤ B(vr) < B(vn)Sk is formed by reflecting Sk−1. That is, Sk−1’s worst vertex vn+1 is

replaced with the reflection point vr.

If B(vr) < B(v1)The objective function is measured at the expansion point defined as:

ve = v − 2 (vn+1 − v) (A.75)

If B(ve) < B(vr)Sk is formed by expanding Sk−1. That is, Sk−1’s worst vertex vn+1 is

replaced with the expansion point ve.

Otherwise

Sk is once again formed by reflecting Sk−1. That is, Sk−1’s worst

vertex vn+1 is replaced with the reflection point vr.

If B(vn) ≤ B(vr) < B(vn+1)The objective function is measured at the outside contraction point de-

fined as:

voc = v − 1

2(vn+1 − v) (A.76)

If B(voc) ≤ B(vr)Sk is formed by outside contracting Sk−1. That is, Sk−1’s worst vertex

vn+1 is replaced with the outside contraction point voc.

Otherwise

Sk is formed by shrinking Sk−1. That is, Sk−1’s best vertex v1 is

kept while all remaining vertices vi [2 ≤ i ≤ (n+1)] are replaced with12(v1 + vi).


If B(vr) ≥ B(vn+1)The objective function is measured at the inside contraction point defined

as:

vic = v + 1

2(vn+1 − v) (A.77)

If B(vic) < B(vn+1)Sk is formed by inside contracting Sk−1. That is, Sk−1’s worst vertex

vn+1 is replaced with the inside contraction point vic.

Otherwise

Sk is once again formed by shrinking Sk−1. That is, Sk−1’s best vertex

v1 is kept while all remaining vertices vi [2 ≤ i ≤ (n+1)] are replacedwith 1

2(v1 + vi).

Several points are worth noting about this optimization algorithm:

• Just like the reflection point (A.74), all of the expansion (A.75), outside con-

traction (A.76) and inside contraction points (A.77) lie along the line inter-

secting v and vn+1 of Sk−1. This makes the updating process, and hence the

entire algorithm, geometrically deterministic.

• Geometric updating of the simplex at each optimization step is a Gradient-Free

process, involving only objective function measurement and comparison. For

this reason, the algorithm is considered highly efficient compared to the Gra-

dient based optimizers discussed previously. The number of objective function

measurements required by each form of geometric update is outlined below:

Geometric Update Objective Function

of Simplex Measurements

Reflection maximum of 2

Expansion 2

Outside Contraction 2

Inside Contraction 2

Shrinkage n+2

Table A.1: Objective function measurements per geometric update


• Whilst technically a local optimization algorithm, a more global emphasis can

be achieved if the initial simplex S0 is defined geometrically large by increasing

d of (A.71). This gives the simplex a chance to explore more of the vector space

before contracting and shrinking.

• After long periods of simplex updating, a situation known as simplex degen-

eration or stagnation may occur whereby vertices of the simplex become ill

aligned (determinant of simplex edge matrix approaches zero) and optimiza-

tion convergence prematurely ceases [133,200]. When this occurs, there is no

choice but to reset the simplex according to (A.71). Practical implementations

of the algorithm take a proactive approach to this problem. They restart the

algorithm on a regular basis and hence deny the simplex any opportunity of

becoming degenerate.

• Convergence theory of this seemingly intuitive algorithm is given in [134,148].

At optimization step k, automated exiting from the algorithm is based on the fol-

lowing mean square criterion [196]:

Υ = ( 1

n + 1)

n+1∑i=1

[B(Sk centroid) −B(Sk vertex i)]2

(A.78)

A large value of Υ generally indicates the simplex is situated on an objective function

slope whilst a small value of Υ generally indicates that the simplex has flattened

out (and contracted) into an objective function minimum region. Based on this

reasoning, if Υ drops below some small predefined value Υ0, the algorithm exits and

returns Sk’s best vertex.

This Nelder-Mead Simplex algorithm is summarized in the flowchart of Fig-

ure A.6. It is also implemented on the laboratory transmitter testbed via the

software template function NelderMeadSimplex(). Corresponding declaration and

definition source code resides in project files NelderMeadSimplexOptimization.h and

NelderMeadSimplexOptimization Templates.cpp respectively. Both files are located


START

get starting point

define initial simplex in terms of & and measure objective function at vertices

set

label vertices of from best to worst

compute , the centroid of 's best vertices

compute reflection point

and measure

is formed by reflecting

compute insidecontraction point

and measure

?

is formed by inside contracting

compute

?OR

‘USER EXIT’ request ?

increment

return best vertex of END

compute expansion point

and measure

is formed by expanding

?

?

?

?

compute outsidecontraction point

and measure

is formed by shrinking

is formed by outside contracting

?

no

yes

no

yes

no

yes

no

yes

no

yes

no

yes

no yes

define initial simplex

form

by either reflecting, contracting, expanding or shrinking

comp

ute characteristics of

hs

hs

S0

k = 1

Sk−

1

Sk−

1

Sk−1

Sk−1

Sk−1

Sk−1

Sk−1

Sk−1

Sk−1 v1 vn+1

v

vr = v − (vn+1 − v)B(vr)

Sk

Sk

Sk

Sk

Sk

Sk

Sk

Υ

n

B(v1) ≤ B(vr) < B(vn)

B(vr) < B(v1)

B(vn) ≤ B(vr) < B(vn+1)

B(vic) < B(vn+1)

B(ve) < B(vr)

B(voc) ≤ B(vr)

Υ < Υ0

k

ve = v − 2 (vn+1 − v)B(ve)

vic = v + 12(vn+1 − v)

B(vic)

voc = v − 12(vn+1 − v)

B(voc)

d

→ B(vr) ≥ B(vn+1)

Figure A.6: Flowchart of Nelder-Mead Simplex optimization algorithm


A.5 Genetic

Rooted in the biological sciences [20, 21, 76, 109], this optimization algorithm ab-

stracts the general principles of evolution (natural selection, reproduction and sur-

vival of the fittest) to generate an estimate of the objective function global mini-

mum [16,17,87]. In keeping with the literature, it is important to note the following

changes in terminology as a result of the algorithm’s biological roots [283,284].

A point in the optimization vector space is now called a chromosome rather than

a variable vector, a single element of a variable vector is now called a gene rather

than a variable and the function to be minimized is now called a fitness function

rather than an objective function. In this sense, a fitter chromosome has a smaller

objective function value.

Figure A.7 outlines the Genetic algorithm in its most basic form. Central to the

algorithm’s operation, the Population Pool represents a set of ξ candidate solutions

to the global minimization problem. After its initial estimation, the Population

Pool is iteratively refined via a biological evolutionary process (involving selection,

reproduction, repopulation) until chromosome concentrations form. These concen-

trations, representing regions of objective function minima, are subsequently used

to estimate the desired global minimum. Such overview concepts are discussed in

more detail in the following.

concentrationsformed?

START

no

yes

END

Population Pool

candidate solutions tothe global minimisation

problem

estimate initial Population Pool

track the formation of chromosome concentrations

estimate global minimum solution from concentration regions

refine Population Pool withimproved candidate solutions viabiological evolutionary process

ξ

Figure A.7: Overview of Genetic optimization algorithm


Estimating the Initial Population Pool

When estimating the initial Population Pool, the goal is not to pinpoint the global

minimum but rather to assemble a set of ξ fit chromosomes from across the entire

vector space. This initial Population Pool merely seeds the algorithm.

In this sense, the simplest method of estimating the initial Population Pool is

to randomly generate χ ≫ ξ chromosomes and measure their fitness. From these χ

chromosomes, the ξ fittest form the Population Pool. An alternative more structured

method is to compute a uniform grid of χ ≫ ξ chromosomes across the vector space

and measure their fitness. Once again, the ξ fittest form the Population Pool.

Refining the Population Pool Via Biological Evolution

The biological evolutionary process used to refine the Population Pool consists of

three steps as outlined in Figure A.8:

Step 1 - Selection The κ fittest chromosomes of the Population Pool are selected

to form the Mating Pool. Chromosomes of the Mating Pool are eligible for

reproduction. This step abstracts the natural selection principle of biological

evolution.

Step 2 - Reproduction � randomly selected pairs of Mating Pool chromosomes

are chosen to mate, with each mating pair reproducing a single chromosome. A

reproduced chromosome is referred to as an offspring whilst each chromosome

of the mating pair is called a parent. The xth gene of an offspring chromosome

is created by:

1. randomly selecting one of the two parent chromosomes and inheriting its

xth gene (a process referred to as crossover).

2. introducing a small random change to this inherited gene (a process re-

ferred to as mutation). In general, the small random change is modeled

with a Normal Distribution.

The entire set of � offspring generated from this reproduction step form the

Offspring Pool. As its name suggests, this step abstracts the reproduction

principle of biological evolution.

Step 3 - Repopulation All � chromosomes of the Offspring Pool are added to the

current Population Pool. This appended Population Pool is then truncated

back to the ξ fittest chromosomes. This step abstracts the survival of the fittest

principle of biological evolution.

Population pool refinement is attributed to the fact that only the fittest chromosomes

are mated during the evolutionary process.



START

no

yes

END

Population Pool


problem

estimate initial Population Pool

track the formation ofchromosome

concentrations

estimate global minimum solution from concentration regions

SELECTION

select fittestchromosomes from

Population Pool to formMating Pool

REPRODUCTION

mate randomlyselected pairs of MatingPool chromosomes toform Offspring Pool

REPOPULATION

add all Offspring Poolchromosomes to

Population Pool andtruncate back to fittest

chromosomes

Mating Pool

chromosomes

OffspringPool

chromosomes

refine Po

pulation P

ool via

biolog

ical evolution

ξ

ξ

κ

κ

�

�

Figure A.8: Biological evolutionary process used to refine the Population Pool

Tracking The Formation of Chromosome Concentrations

Repeated evolutionary refinement of the Population Pool causes chromosomes to

become fitter and hence gradually concentrate in regions of objective function min-

ima. Mathematically speaking, a chromosome concentration is defined as a set of

chromosomes exhibiting similar individual gene values. By searching for similarities

in gene values, pattern recognition algorithms can subsequently identify and track

the formation of individual chromosome concentrations.

The chromosome concentration formation process can be demonstrated visually,

as follows, via indicative gene plots of the Population Pool after short, medium and

long-term refinement.


After short-term refinement (Figure A.9a), the Population Pool’s chromosomes

are still fairly raw and scattered throughout the entire vector space. As a re-

sult, a plot of the Population Pool appears random with no sign of chromosome

concentrations.

After medium-term refinement (Figure A.9b), the Population Pool’s chromo-

somes are substantially refined with distinct chromosome concentrations form-

ing. One concentration represents the global minimum region whilst the others

represent local minima regions. In the example of Figure A.9b, three concen-

trations exist.

After long-term refinement (Figure A.9c), the Population Pool is refined to

the point where all chromosomes concentrate around the global minimum.

Evolutionary refinement has effectively run to completion. Those previous

medium-term concentrations representing local minima regions have gradually

dispersed (in order of least fit) with their chromosomes reconcentrating around

this global minimum.

Ideally, with the above in mind, the goal would be to run evolutionary refinement

to completion (long-term refinement) such that a single global minimum region

is identified. This is impractical however given finite time constraints. Practical

implementations of the algorithm therefore halt refinement once a small number ϑ

of distinct chromosome concentrations have formed (medium-term refinement) and

estimate the desired global minimum from these ϑ concentrations via regional local

optimization and comparison as discussed in the next section.

Estimating the Global Minimum From the Concentration Regions

Estimating the desired global minimum from the small number ϑ of distinct chro-

mosome concentrations is a three step process:

1. The centroid of each chromosome concentration is computed.

2. A local optimizer is employed at each computed centroid to locate the corre-

sponding local minimum.

3. The desired global minimum is estimated as the smallest of these ϑ local minima.

This Genetic algorithm is summarized in the flowchart of Figure A.10. It is

also implemented on the laboratory transmitter testbed via the template function

Genetic(). Corresponding declaration and definition source code resides in the

project files GeneticOptimization.h and GeneticOptimization Templates.cpp respec-

tively. Both files are located within folder Software\Cpp\ on the accompanying DVD.

Gene 1 Gene 2 Gene 3 Gene nChromosome

Genes

Gen

e V

alu

es

LEGEND

Chromosome 1Chromosome 2Chromosome 3

Chromosome ξ

(a)


Genes

Gen

e V

alu

es

LEGEND


Chromosome ξ

(b)


Genes

Gen

e V

alu

es

LEGEND


Chromosome ξ

(c)

Figure A.9: Gene plots of the Population Pool after (a) short-term (b) medium-termand (c) long-term refinement


START

no

yes

END

Population Pool


problem

randomly generate chromosomes across vector

space & measure their fitness

for each of the concentrations …compute centroid

SELECTION

select fittestchromosomes from

Population Pool to formMating Pool

REPRODUCTION

mate randomlyselected pairs of MatingPool chromosomes toform Offspring Pool

REPOPULATION

add all Offspring Poolchromosomes to

Population Pool andtruncate back to fittest

chromosomes

Mating Pool chromosomes

OffspringPool

chromosomes

refine Po

pulation P

ool via

biolog

ical evolution

search the Population Poolfor chromosome gene value

similarities and henceidentify distinct chromosome

concentrations

estimate initial P

opula

tion Poo

ltrack the

forma

tion ofchro

mosom

e concentra

tions

from these chromosomes,select fittest to form initial

Population Pool

estimate g

lobal m

inimum

solution from co

ncentration regio

ns

for each of the centroids …employ local optimizer to findcorresponding local minimum

estimate global minimum as thesmallest of these local minima

χ

χ ≫ ξ

ϑ

ϑ

ϑ

ϑ

ξ

ξ

ξ

κ

κ

�

�

Figure A.10: Flowchart of Genetic optimization algorithm

Appendix B

Mathematical Derivations &

Formulas

B.1 Estimation of the Gradient Vector g

B.1.1 Finite Differences

In cases when the objective function B(h) exhibits minimal randomness, the Gra-

dient vector g:

g ≜ ∂B(h)∂h

∣hk

=⎡⎢⎢⎢⎢⎣

∂B(h)∂h1

∂B(h)∂h2

⋯ ∂B(h)∂hn

⎤⎥⎥⎥⎥⎦

T ???????????hk

(B.1)

can be estimated numerically via the traditional Finite Differences technique. In

the forward-difference version of the technique, the ith element of g is estimated as:

∂B(h)∂hi

???????????hk

≈ B(hk + εei) −B(hk)ε

(B.2)

where ε is a small positive scalar and ei is the unit vector along the ith dimension of

the vector space h. In total, (n + 1) objective function measurements are required

to compute this estimate.

In the more accurate central-difference version of the technique, the ith element of

g is estimated as:

∂B(h)∂hi

???????????hk

≈ B(hk + εei) −B(hk − εei)2ε

(B.3)

With this more accurate estimate comes the need for a greater number of objective

function measurements, specifically 2n in total. In both versions of the technique,

206

APPENDIX B. MATHEMATICAL DERIVATIONS & FORMULAS 207

ε is chosen as small as possible without becoming affected by the finite arithmetic

errors introduced by the computer [54].

B.1.2 Least Squares

In cases when the objective function B(h) exhibits substantial randomness (mea-

surement noise, stochastic variance), the previous Finite Differences estimation tech-

nique becomes unreliable and must be abandoned for this more robust but less ef-

ficient Least Squares estimation technique. It is noted that this technique is not

borrowed from the literature but rather developed here in direct response to the

needs of this research.

Let the objective function in the vicinity of hk be estimated in terms of a linear

parametric model as follows:

B(hk + d ) ≈ c + dTa (B.4)

Given a set ofm objective function measurements taken uniformly within the vicinity

of hk:

{ B(hk + d1 ), B(hk + d2 ), ⋯ , B(hk + dm ) } (B.5)

the goal is to compute the parameters c and a of the linear model so as to minimize

the mean squared estimation error. That is, c and a are computed as:

argminc,a

⎛⎝Avi[{B(hk + di ) − (c + dT

ia) }2]⎞⎠

(B.6)

where Av[ ⋅ ] is the average operator. This specific value of the parameter a is then

taken to be the desired estimate of the objective function Gradient vector g. Three

points are worth noting:

• Quality of the Gradient vector estimate is directly determined by the choice

of objective function measurement points in Equation (B.5). Highest quality

is achieved when points are chosen uniformly from within a small boxed

region [ (hk − ε) ≤ h ≤ (hk + ε) ] surrounding hk. In this context, one strategy

is to partition the small boxed region into a uniform grid of measurement

points. An alternative, less structured strategy is to perform a large number of

random selections from within the small boxed region, yielding a near-uniform

distribution of measurement points. A hybrid of the above two strategies could

also be used with good effect.

In all of the above strategies for choosing objective function measurement

points, the number of points chosen should be substantial. The greater the

number of measurement points, the more robust this Least Squares estimation


technique is against objective function randomness. A safe recommendation

is at least 3n points.

• Since the mean squared estimation error of (B.6) is derived from the set of ob-

jective function measurement points in (B.5), it is more appropriate to discuss

this Least Squares technique in terms of set averages Av[ ⋅ ] rather than prob-

abilistic expectations E[ ⋅ ]. Irrespective, the mathematical theory is identical

in both cases.

• To obtain the optimal linear model in the presence of substantial objective

function randomness, both parameters c and a must be included in the mini-

mization of (B.6). Choosing c deterministically, instead of including it in the

minimization, leads to a degraded estimate of g.

We begin by rewriting the linear parametric model of (B.4) more compactly as:

c + dTa = dTw (B.7)

where:

d =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

d1

d2

⋮dn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

a =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

a1

a2

⋮an

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

d =⎡⎢⎢⎢⎢⎣

d

1

⎤⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

d1

d2

⋮dn

1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

w =⎡⎢⎢⎢⎢⎣

a

c

⎤⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

a1

a2

⋮an

c

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B.8)

Substituting this compact form into (B.6) and replacing the ‘argmin’ operator sub-

scripts (c, a) with w gives:

argminw

⎛⎝Avi[{B(hk + di ) − d

T

iw }2]⎞⎠

(B.9)

Expanding the inner squared term of (B.9) gives:

argminw

⎛⎝Avi[B(hk + di )2 + wTdi d

Tiw − 2B(hk + di ) d

Tiw ]

⎞⎠

(B.10)

From the linearity property of the Av[ ⋅ ] operator, (B.10) can be rewritten as:

argminw

⎛⎝Avi[B(hk+di )2] + wTAv

i[di d

T

i ]w − 2Avi[B(hk+di ) d

T

i ]w⎞⎠

(B.11)


Let:

C = Avi[B(hk + di )2] (B.12)

R = Avi[di d

T

i ] (B.13)

P = Avi[B(hk + di ) d

T

i ] (B.14)

(B.11) can then be rewritten compactly as:

argminw

⎛⎝C + wTRw − 2P w

⎞⎠

(B.15)

The bracketed term of (B.15) is recognized as a convex quadratic function of w.

Convexity results from the fact that R is positive-definite; that is, given a set of di

chosen uniformly in the vicinity of hk:

wTRw = wTAvi[di d

Ti ]w (B.16)

= Avi[wTdi d

Tiw] (B.17)

= Avi[wTdiw

Tdi] (B.18)

= Avi[ (wTdi)

2] > 0 ∀ w ≠ 0 (B.19)

It follows that (B.15) has a unique solution of w existing where the Gradient of the

quadratic function equals zero:

∂

∂w( C + wTRw − 2P w ) = 0 (B.20)

2Rw − 2P = 0 (B.21)

Rw = P (B.22)

w = R−1P (B.23)

Mindful of matrix conditioning effects, w is computed via the (B.22) linear system

of equations (LU decomposition of R) instead of the (B.23) R inversion [213]. From

the definition of w given in (B.8), the desired a and therefore g estimate is simply

the first n elements of w.


B.2 Estimation of the Hessian Matrix H

B.2.1 Finite Differences

In cases when the objective function B(h) exhibits minimal randomness, the Hessian

matrix H :

H ≜ ∂

∂h(∂B(h)

∂h)T

∣hk

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂2B(h)∂h1 ∂h1

∂2B(h)∂h1 ∂h2

⋯ ∂2B(h)∂h1 ∂hn

∂2B(h)∂h2 ∂h1

∂2B(h)∂h2 ∂h2

⋯ ∂2B(h)∂h2 ∂hn

⋮ ⋮ ⋱ ⋮

∂2B(h)∂hn ∂h1

∂2B(h)∂hn ∂h2


⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

???????????????????????????????????????????????????hk

(B.24)

can be estimated numerically via the same Finite Differences concepts first intro-

duced in Appendix B.1.1 for estimating the Gradient vector g. Knowledge of these

concepts is assumed in the following discussion.

Let the (i, j)th element of H be rewritten as:

∂2B(h)∂hi ∂hj

∣hk

=⎡⎢⎢⎢⎢⎣

∂

∂hi(∂B(h)

∂hj)⎤⎥⎥⎥⎥⎦

????????????hk

(B.25)

If the round bracketed term of (B.25) is interpreted simply as a function of h then

the right hand side can be estimated in terms of forward differences:

∂2B(h)∂hi ∂hj

∣hk

≈(∂B(h)

∂hj)∣

hk+εei

− (∂B(h)∂hj

)∣hk

ε(B.26)

where ε is a small positive scalar and ei is the unit vector along the ith dimension

of the vector space h. It is worth noting that a more accurate central difference

estimate could be used here however practical implementations generally favor the

forward difference estimate due to its greater efficiency in the long run.


Estimating the numerator partial derivatives of (B.26), again in terms of forward

differences, gives:

∂2B(h)∂hi ∂hj

∣hk

≈

⎛⎝B(hk + εei + εej) −B(hk + εei)

ε

⎞⎠−⎛⎝B(hk + εej) −B(hk)

ε

⎞⎠

ε(B.27)

Rearranging (B.27) then gives the desired estimate of the (i, j)th element of H in

terms of objective function measurements:

∂2B(h)∂hi ∂hj

∣hk

≈ B(hk + εei + εej) −B(hk + εei) −B(hk + εej) +B(hk)ε2

(B.28)

If the objective function measurements in (B.28) are stored in memory once ac-

quired and appropriately recycled for all elements of the Hessian matrix, a total of

(n2−n2 + 2n + 1) objective function measurements are required to form the H esti-

mate.

B.2.2 Hybridization of Finite Differences & Least Squares

In cases when the objective function B(h) exhibits substantial randomness (mea-

surement noise, stochastic variance), partial derivatives of B(h) cannot be reliably

estimated via Finite Differences. In the context of the previous section, this means

the step from (B.26) to (B.27) falls down and leads to a generally poor estimate in

(B.28).

This problem can be avoided altogether however by row vectorizing the estima-

tion of H (estimating H row-by-row instead of element-by-element), in which case

partial derivatives of B(h) no longer appear individually but rather are grouped into

Gradient vectors which can be robustly estimated by the Least Squares technique

of Appendix B.1.2. The Hessian estimation technique thus born and presented here

is more robust in the presence of objective function randomness but expectedly less

efficient.

As will become evident below, this robust estimation technique can be consid-

ered a hybrid technique since rows of H are initially estimated in terms of Finite

Differences whilst subsequent partial derivatives of B(h) are estimated in terms of

Least Squares.


With the Hessian matrix H defined as:

H ≜ ∂

∂h(∂B(h)

∂h)T

∣hk

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂2B(h)∂h1 ∂h1

∂2B(h)∂h1 ∂h2

⋯ ∂2B(h)∂h1 ∂hn

∂2B(h)∂h2 ∂h1

∂2B(h)∂h2 ∂h2

⋯ ∂2B(h)∂h2 ∂hn

⋮ ⋮ ⋱ ⋮

∂2B(h)∂hn ∂h1

∂2B(h)∂hn ∂h2


⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

???????????????????????????????????????????????????hk

(B.29)

the ith row of H can be expressed as:

H i,∀ = ∂

∂hi(∂B(h)

∂h)T ???????????hk

(B.30)

Estimating the outer partial derivative of (B.30) via forward differences then gives:

H i,∀ =(∂B(h)

∂h)T

∣hk+εei

− (∂B(h)∂h

)T

∣hk

ε(B.31)

where ε is a small positive scalar and ei is the unit vector along the ith dimension of

the vector space h. It is worth noting that (B.30) and (B.31) are the row vectorized

extension of (B.25) and (B.26) respectively of the previous section.

Since the left and right numerator terms in (B.31) are recognized as the Gradient

vector of B(h) at (hk+εei) and hk respectively, each can be estimated via the robust

Least Squares technique of Appendix B.1.2. With Appendix B.1.2 recommending

at least 3n objective function measurements per Gradient estimate, it follows that

a total of at least 3n2 + 3n objective function measurements are required to form

the entire H estimate, assuming each Gradient estimate is computed independently

and the right numerator term of (B.31) is recycled for each row.


B.3 Eigen Properties of the Hessian Matrix H

Presented below are the eigen properties of the Hessian matrixH that are specifically

utilized in the theory of the Trust Region Newton and Alpha Branch & Bound

optimization algorithms:

Property 1: Since H is a square (n × n), real, symmetric matrix [118]:

• all eigenvalues λ1 ≤ λ2 ≤ ⋯ ≤ λn are real

• a full set of n linearly independent and orthonormal eigenvectors

q1, q2, ⋯ , qn exist whereby qTi qj = 0 for i ≠ j and qT

i qj = 1 for i = j

Property 2: As a consequence of Property 1, H can be written in diagonal form [11]

as:

H = QΛQT (B.32)

where Λ is the diagonal spectral matrix:

Λ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

λ1 0 ⋯ 0

0 λ2 ⋯ 0

⋮ ⋮ ⋱ ⋮0 0 ⋯ λn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B.33)

and Q is the orthogonal modal matrix:

Q = [q1 q2 ⋯ qn] (B.34)

The orthogonality of Q stems from the fact [143]:

QTQ = I = Q−1Q and therefore QT = Q−1 (B.35)

Property 3: Geometrically, the eigenvectors of H define the principle axes of the

quadratic surface dTHd whilst the eigenvalues of H represent the second

order derivatives of dTHd with respect to the principle axes [285]. A principle

axis of dTHd is defined as the set of linear points Cq for which the gradient

is collinear with the origin. A principle axis can be visualized as passing

orthogonally through the hyper-ellipsoidal contours of the dTHd surface. This

same geometrical interpretation can be applied to the full quadratic model of

(A.10), only this time the quadratic surface will be translated in the d vector

space by the inclusion of the constant and linear terms.


Property 4: It can be shown mathematically [118] or interpreted geometrically

from Property 3 (eigenvalues represent second order derivatives) that the eigen-

values of H determine the definiteness of the quadratic term dTHd:

• If all eigenvalues of H are positive than dTHd > 0 for all d ≠ 0 and H is

called positive-definite.

• If all eigenvalues of H are non-negative and at least one of the eigenvalues

is zero then dTHd ≥ 0 for all d ≠ 0 and H is called positive-semidefinite.

• If all eigenvalues of H are negative than dTHd < 0 for all d ≠ 0 and H

is called negative-definite.

• If all eigenvalues of H are non-positive and at least one of the eigenvalues

is zero then dTHd ≤ 0 for all d ≠ 0 and H is called negative-semidefinite.

• If the eigenvalues of H are both positive and negative than dTHd takes

at least one positive value and at least one negative value and H is called

indefinite.

Property 5: The eigenvalues of [H + λI] are (λ1 + λ, λ2 + λ, ⋯ , λn + λ) whilst

the eigenvectors of [H + λI] are the same as the eigenvectors of H [275]. It

follows that weighting the entire diagonal of H by a constant simply changes

the second order characteristics along the principle axes.

Property 6: The determinant of H can be derived from its eigenvalues:

det (H) =n

∏i=1

λi (B.36)

If det (H) ≠ 0 then rank (H) = n and H is nonsingular. In this case the linear

system of equations Hx = b has a unique solution.

If det (H) = 0 then rank (H) < n and H is singular. In this case, the linear

system of equations Hx = b has either a non-unique solution or no solution

exists.

Property 7: Let Rg be a general boxed region of vector space h with domain:

[hL,Rg ≤ h ≤ hU,Rg] (B.37)

where hL,Rg and hU,Rg represent the lower and upper constraints of region Rg

respectively. Within region Rg, it is assumed that the Hessian matrix of the

function of h is itself dependent on h and therefore represented as:


H(h) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

H1,1(h) H1,2(h) ⋯ H1,n(h)H2,1(h) H2,2(h) ⋯ H2,n(h)

⋮ ⋮ ⋱ ⋮Hn,1(h) Hn,2(h) ⋯ Hn,n(h)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B.38)

If:

Hi,j = maxhL,Rg ≤h≤hU,Rg

Hi,j(h) (B.39)

and

¯Hi,j = min

hL,Rg ≤h≤hU,RgHi,j(h) (B.40)

then the interval Hessian matrix of region Rg is defined as:

[H]Rg =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

[¯H1,1, H1,1] [

¯H1,2, H1,2] ⋯ [

¯H1,n, H1,n]

[¯H2,1, H2,1] [

¯H2,2, H2,2] ⋯ [

¯H2,n, H2,n]

⋮ ⋮ ⋱ ⋮

[¯Hn,1, Hn,1] [

¯Hn,2, Hn,2] ⋯ [

¯Hn,n, Hn,n]

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B.41)

where [¯Hi,j, Hi,j] represents the closed interval

¯Hi,j ≤ Hi,j ≤ Hi,j. (B.41) can

be written in shorthand as:

[H]Rg = [¯H , H] (B.42)

where:

¯H =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

¯H1,1

¯H1,2 ⋯

¯H1,n

¯H2,1

¯H2,2 ⋯

¯H2,n

⋮ ⋮ ⋱ ⋮

¯Hn,1

¯Hn,2 ⋯

¯Hn,n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B.43)


and

H =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

H1,1 H1,2 ⋯ H1,n

H2,1 H2,2 ⋯ H2,n

⋮ ⋮ ⋱ ⋮Hn,1 Hn,2 ⋯ Hn,n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B.44)

The entire set of matrices represented by the interval bounds of [H]Rg is re-

ferred to as the [H]Rg interval Hessian matrix family { [H]Rg}. By definition:

{H(h) ∀ [hL,Rg ≤ h ≤ hU,Rg] } ⊆ { [H]Rg} (B.45)

Property 8: Let H be the arbitrary (not necessarily Hessian) square matrix:

H =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

H1,1 H1,2 ⋯ H1,n

H2,1 H2,2 ⋯ H2,n

⋮ ⋮ ⋱ ⋮Hn,1 Hn,2 ⋯ Hn,n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B.46)

From matrix H, n complex-z-plane disks Di can be defined as:

∣ z −Hi,i ∣ ≤n

∑j=1j≠i

∣Hi,j ∣ for integer 1 ≤ i ≤ n (B.47)

It is seen that each disk Di is centered at diagonal element Hi,i with ra-

dius equal to the sum of the magnitudes of all row i off-diagonal elements.

Gerschgorin’s Theorem [118] then states that each disk Di represents the en-

tire bounds of one of the eigenvalues of H ; n disks, n eigenvalues. It is noted

that in this theorem, each eigenvalue is counted with its algebraic multiplicity.

For the special case when H is a real symmetric matrix (Hessian), all eigen-

values must be real according to the previous Property 1. It follows that the

complex bounding region of each disk can be reduced to a real bounding in-

terval equal to that part of the real number line enclosed by the disk. It is

worth noting that since the matrix is real in this case, the center of each disk

(Hi,i) will lie on the real number line.

From the above discussion, it logically follows that diagonally weighting a

square matrix has the effect of translating its Gerschgorin disks and therefore

its eigenvalue bounds.

Appendix C

Patent

The full document set representing the patent description, drawings, claims, ap-

plication filings and international search report is included on the accompanying

DVD within the Patent folder. For brevity, only patent claims are included in this

appendix. These claims represent the scope of the IP generated from this research.

217

CLAIMS

1. A method for linearising a multi-carrier radio frequency transmitter or a

multi-user CDMA radio frequency transmitter, including the steps of:

measuring a function of out-of-band signal power in the frequency

domain at an output of the radio frequency transmitter; and 5

applying digital base-band pre-distortion to the radio frequency

transmitter according to the measured function of the out-of-band signal

power;

wherein the digital base-band pre-distortion is performed by a digital

base-band pre-distortion network. 10

2. The method of claim 1 wherein digital base-band pre-distortion

network coefficients of the digital base-band pre-distortion network are

optimised to minimise the measured function of the out-of-band signal power.

15

3. The method of claim 2 wherein the digital base-band pre-distortion

network coefficients are optimised whilst the transmitter is broadcasting.


network is a non-linear behavioural model with memory. 20

5. The method of claim 4 wherein the non-linear behavioural model with

memory is a pruned Volterra Series.

6. The method of claim 2 wherein the digital base-band pre-distortion 25

network coefficients are pruned Volterra Series kernel coefficients.


network is given by the equation:

21

1

11

0

2)1(212 ][][][][][][

P

a

RM

k

aa Rknxnxnxkhnxny

where ][12 kh a are the digital base-band pre-distortion network kernel

coefficients.

5

8. The method of claim 7 wherein the memory length M is estimated by:

a) pruning the digital base-band pre-distortion network to a 3rd

order single delay digital base-band pre-distortion network given

by the equation: 2

3 ][][][][][ knxnxkhnxny10

b) Sweeping a delay variable ( k ) of the 3rd order single delay pre-

distortion network from zero upwards; and

c) Observing a value of k when an asymmetry of the transmitter

output adjacent channel power spectrum changes wherein the value

of k is equal to the memory length M .15

9. The method of claim 1 wherein the function of the out-of-band signal

power is a measure of transmitter output non-linearity.

10. The method of claim 9 wherein the function of the out-of-band signal 20

power involves accumulating a weighted out-of-band power spectral density

with respect to frequency.

11. The method of claim 10 wherein the function of the out-of-band signal

power is given by the equation: 25

fffPSDfWfPSDfWWACP

UACLAC)()()()(

12. The method of claim 11 wherein the weighting function )( fW , for

either the lower adjacent channel (LAC) or upper adjacent channel (UAC), is

a non-increasing function of Iff .5

13. The method of claim 10 or claim 11 wherein the power spectral

density is measured with a spectrum analyser.

14. The method of claim 7 wherein a subset of the digital base-band pre-10

distortion network kernel coefficients is optimised separately.

15. The method of claim 14 wherein a combination of 3rd order, a

combination of 3rd and 5th order or a combination of 3rd and 5th and 7th order

digital base-band pre-distortion network kernel coefficients is optimised 15

separately.


network kernel coefficients are optimised according to a local minimum non-

gradient based algorithm. 20


network kernel coefficients are optimised according to a global minimum non-

gradient based algorithm.

25

18. The method of claim 16 wherein the local minimum non-gradient

based algorithm is a Nelder-Mead Simplex algorithm.

19. The method of claim 17 wherein the global minimum non-gradient

based algorithm is a Genetic algorithm.

20. The method of claim 14 wherein a subset of the digital base-band pre-5

distortion network kernel coefficients, all of the same non-linear order, is

optimised separately according to a gradient based algorithm.

21. The method of claim 20 wherein the gradient based algorithm is a

local minimum Gradient Descent algorithm. 10

Appendix D

Publications

The following journal articles are products of this research activity and are included

in this appendix in respective order:

B. D. Laki and C. J. Kikkert, “Adaptive Digital Predistortion For Wideband High

Crest Factor Applications Based On The WACP Optimization Objective: A

Conceptual Overview,” IEEE Transactions On Broadcasting, Vol. 58, No. 4,

pp. 609–618, December 2012.

B. D. Laki and C. J. Kikkert, “Adaptive Digital Predistortion For Wideband High

Crest Factor Applications Based On The WACP Optimization Objective: An

Extended Analysis,” IEEE Transactions On Broadcasting, Vol. 59, No. 1, pp.

136–145, March 2013.

222

IEEE TRANSACTIONS ON BROADCASTING, VOL.58, NO. 4, DECEMBER 2012 609

Adaptive Digital Predistortion for Wideband HighCrest Factor Applications Based on the WACP

Optimization Objective: A Conceptual OverviewBradley Dean Laki, Graduate Member, IEEE, and Cornelis Jan Kikkert, Life Senior Member, IEEE

Abstract—This paper proposes a method of digital predistor-tion suitable for wideband high crest factor applications such asthose encountered in DAB, DVB-T and WCDMA transmitters.The proposed method is advantageous for four main reasons.Firstly, it utilizes a reliable frequency domain measure of trans-mitter output nonlinearity, specifically the Weighted AdjacentChannel Power (WACP), as the objective for predistortion filterparameter estimation. This is in direct contrast to traditionalapproaches which utilize a time domain measure obtained viaa full feedback path and potentially corrupted by gain andphase compensation error as well as ADC distortion. Secondly,the method models predistortion filter parameter estimation asa generic nonlinear mathematical optimisation problem. Thismodel assumes a nonconvex objective function and thereforeutilizes both global and local optimization algorithms to achievetrue global convergence. This is once again in direct contrastto traditional approaches which model predistortion filter pa-rameter estimation as a linear regression problem. Such amodel incorrectly assumes a convex error surface and thereforerestricts itself to inadequate local optimisation algorithms whichunfortunately cannot guarantee true global convergence. Thirdly,the method’s predistortion filter is a pruned Volterra Series withmemory which utilizes a hybrid pruning strategy in order to keephigh order kernels to a practically manageable size, suitable foroptimization parameter estimation. Predistortion filter memoryultimately makes the method highly suited to wideband applica-tions. Finally, predistortion filter parameter estimation does notrequire known test signals to be injected into the transmitterand therefore the technique is on-air adaptive. This means anytransmitter using this method of digital predistortion will be bothon-air and optimally linearized for its entire operational life.

Preliminary results obtained from actual hardware are pre-sented.

Index Terms—Adjacent channel power, CDMA, DAB, DVB-T,Linearization techniques, Nonlinear distortion, OFDM, Opti-mization, Power amplifiers, Predistortion, Radio transmitters,Volterra Series

IEEE TRANSACTIONS ON BROADCASTING, VOL.59, NO. 1, MARCH 2013 136

Adaptive Digital Predistortion for Wideband HighCrest Factor Applications Based on the WACPOptimization Objective: An Extended Analysis

Bradley Dean Laki, Graduate Member, IEEE, and Cornelis Jan Kikkert, Life Senior Member, IEEE

Abstract—This paper provides an extended analysis of theadaptive digital predistortion technique initially proposed andconceptually overviewed in [1]. This digital predistortion tech-nique is suitable for wideband high crest factor applications(DAB, DVB-T & WCDMA high power transmitters) and over-comes the technical deficiencies of the traditional Direct Learningmethod. Specifically, predistortion filter parameter estimation ismodeled as a generic mathematical optimization problem insteadof a linear regression problem. In addition, the optimizationobjective is derived in the frequency rather than time domain. Ahybridly pruned Volterra Series with memory is used to imple-ment the predistortion filter. Hybrid pruning leads to a smalloptimization vector space whilst predistortion filter memorymakes the method well suited to wideband applications. Giventhat predistortion filter parameter estimation does not rely onknown test signals being injected into the transmitter, the methodis on-air adaptive. Implementation aspects of the techniquenot covered in [1] but requiring extended coverage in thispaper include selection of mathematical optimization algorithms,predistortion filter memory estimation, WACP weighting functionapplication and on-air adaption performance. Results obtainedfrom actual hardware are presented.

Index Terms—Adjacent channel power, CDMA, DAB, DVB-T,Linearization techniques, Nonlinear distortion, OFDM, Opti-mization, Power amplifiers, Predistortion, Radio transmitters,Volterra Series

Appendix E

Data Sheets

Data sheets for the laboratory transmitter testbed driver and power amplifiers are

included in this appendix in respective order.

243

Surface Mount

Monolithic Amplifier

Page 1 of 4

ISO 9001 ISO 14001 AS 9100 CERTIFIEDMini-Circuits®

P.O. Box 350166, Brooklyn, New York 11235-0003 (718) 934-4500 Fax (718) 332-4661 The Design Engineers Search Engine Provides ACTUAL Data Instantly at®

Notes: 1. Performance and quality attributes and conditions not expressly stated in this specification sheet are intended to be excluded and do not form a part of this specification sheet. 2. Electrical specificationsand performance data contained herein are based on Mini-Circuit’s applicable established test performance criteria and measurement instructions. 3. The parts covered by this specification sheet are subject toMini-Circuits standard limited warranty and terms and conditions (collectively, “Standard Terms”); Purchasers of this part are entitled to the rights and benefits contained therein. For a full statement of the StandardTerms and the exclusive rights and remedies thereunder, please visit Mini-Circuits’ website at www.minicircuits.com/MCLStore/terms.jsp.

For detailed performance specs& shopping online see web site

minicircuits.comIF/RF MICROWAVE COMPONENTS

simplified schematic and pin description

Function Pin Number Description

RF IN 1RF input pin. This pin requires the use of an external DC blocking capacitor chosen for the frequency of operation.

RF-OUT and DC-IN 3

RF output and bias pin. DC voltage is present on this pin; therefore a DC blocking capacitor is necessary for proper operation. An RF choke is needed to feed DC bias without loss of RF signal due to the bias connection, as shown in “Recommended Application Circuit”.

GND 2,4Connections to ground. Use via holes as shown in “Suggested Layout for PCB Design” to reduce ground path inductance for best performance.

General DescriptionGali 52+ (RoHS compliant) is a wideband amplifier offering high dynamic range. Lead finish is SnAgNi. It has repeatable performance from lot to lot, and is enclosed in a SOT-89 package. It uses patented Tran-sient Protected Darlington configuration and is fabricated using InGaP HBT technology. Expected MTBF is 14,000 years at 85°C case temperature. Gali 52+ is designed to be rugged for ESD and supply switch-on transients.

GROUND

RF IN

RF-OUT and DC-IN

REV. RM120653D60129EE-7974QGALI-52+RS/YB/FL100830

DC-2 GHz

CASE STYLE: DF782PRICE: $1.29 ea. QTY. (30)

Gali 52+

+ RoHS compliant in accordance with EU Directive (2002/95/EC)

The +Suffix has been added in order to identify RoHS Compliance. See our web site for RoHS Compliance methodologies and qualifications.

Features

Applications

3 RF-OUT & DC-IN

2 GROUND

1 RF-IN

4

Monolithic InGaP HBT MMIC Amplifier

ISO 9001 ISO 14001 AS 9100 CERTIFIEDMini-Circuits®

P.O. Box 350166, Brooklyn, New York 11235-0003 (718) 934-4500 Fax (718) 332-4661 The Design Engineers Search Engine Provides ACTUAL Data Instantly at®

Notes: 1. Performance and quality attributes and conditions not expressly stated in this specification sheet are intended to be excluded and do not form a part of this specification sheet. 2. Electrical specificationsand performance data contained herein are based on Mini-Circuit’s applicable established test performance criteria and measurement instructions. 3. The parts covered by this specification sheet are subject toMini-Circuits standard limited warranty and terms and conditions (collectively, “Standard Terms”); Purchasers of this part are entitled to the rights and benefits contained therein. For a full statement of the StandardTerms and the exclusive rights and remedies thereunder, please visit Mini-Circuits’ website at www.minicircuits.com/MCLStore/terms.jsp.

For detailed performance specs& shopping online see web site

minicircuits.comIF/RF MICROWAVE COMPONENTS

Page 2 of 4

Electrical Specifications at 25°C and 50mA, unless notedParameter Min. Typ. Max. Units

Frequency Range* DC 2

Gain 22.9 dB

20.8

16 17.8

15.9

14.4

Input Return Loss 16.5 dB

Output Return Loss 15.5 dB

Output Power @ 1 dB compression 13.5 15.5 dBm

Output IP3 32 dBm

Noise Figure 2.7 dB

Recommended Device Operating Current 50 mA

Device Operating Voltage 4.0 4.4 4.8 V

Device Voltage Variation vs. Temperature at 50 mA -3.2 mV/°C

Device Voltage Variation vs. Current at 25°C 3.5 mV/mA

Thermal Resistance, junction-to-case1 85 °C/W

Note: Permanent damage may occur if any of these limits are exceeded. These ratings are not intended for continuous normal operation.1Case is defined as ground leads.*Based on typical case temperature rise 3°C above ambient.

Absolute Maximum Ratings

Gali 52+

Parameter Ratings

Operating Temperature* -45°C to 85°C

Storage Temperature -65°C to 150°C

Operating Current 65mA

Input Power 13dBm

*

MODEL 530303820 - 1000 MHz

25 WATTS LINEAR POWER RF AMPLIFIER

Solid StateBroadband High

Power RF Amplifier

The 5303038 is a 25 Watt broadband amplifier that covers the 20 – 1000 MHz frequency range. This small and lightweight amplifier utilizes Class A/AB linear power devices that provide an excellent 3rd order intercept point, high gain, and a wide dynamic range.

Due to robust engineering and employment of the most advanced devices and components, this amplifier achieves high efficiency operation with proven reliability. Like all OPHIRRF

amplifiers, the 5303038 comes with an extended multiyear warranty.

5300 Beethoven Street, Los Angeles, CA 90066 TEL: (310)306-5556 FAX: (310)821-7413

WEB: www.ophirrf.com E-MAIL: [email protected]

Parameter Specification @ 25° CElectrical

1 Frequency Range 20 – 1000 MHz 2 Saturated Output Power 25 Watts typical 3 Power Output @ 1dB Comp. 10 Watts min 4 Small Signal Gain +46 dB min 5 Gain Flatness + 1.5 dB max 6 IP3 +48 dBm typical 7 Input VSWR 2:1 max 8 Harmonics -20 dBc typical @ 10 Watts 9 Spurious Signals < -60 dBc typical @ 10 Watts

10 Input/Output Impedance 50 Ohms nominal 11 DC Input Current 4.5 Amps max 12 DC Input 28 VDC nominal 13 RF Input 0 dBm max 14 RF Input Signal Format CW/AM/FM/PM/Pulse 15 Class of Operation AB

Mechanical16 Dimensions 6” x 3” x 1.1” 17 Weight 2 lb. max 18 Connectors SMA female 19 Grounding Chassis 20 Cooling Adequate Heatsink Required

Environmental21 Operating Temperature 0º C to +50º C 22 Operating Humidity 95% Non-condensing 23 Operating Altitude Up to 10,000’ Above Sea Level 24 Shock and Vibration Normal Truck Transport

Specifications subject to change without notice.

Approved By: ________________________________________________ Date: _____________________ 04/09

Solid State RF Amplifier with Push-Pull Circuitry

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